System and method for determining seasonal energy consumption with the aid of a digital computer

ABSTRACT

A system and method for determining seasonal energy consumption with the aid of a digital computer is provided. Through a power metering energy loads for a building situated in a known location are assessed as measured over a seasonal time period. Outdoor temperatures for the building are assessed as measured over the seasonal time period through a temperature monitoring infrastructure. A digital computer comprising a processor and a memory that is adapted to store program instructions for execution by the processor is operated, the program instructions capable of: expressing each energy load as a function of the outdoor temperature measured at the same time of the seasonal time period in point-intercept form; and taking a slope of the point-intercept form as the fuel rate of energy consumption during the seasonal time period.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application is a continuation of U.S. patent U.S. Pat. No.10,797,639, issued Sep. 16, 2020, which is a continuation-in-part ofU.S. Pat. No. 10,651,788, issued May 12, 2020; which is a continuationof U.S. Pat. No. 9,880,230, issued Jan. 30, 2018; which is acontinuation-in-part of U.S. Pat. No. 8,682,585, issued Mar. 25, 2014;which is a continuation of U.S. Pat. No. 8,437,959, issued May 7, 2013,pending; which is a continuation of U.S. Pat. No. 8,335,649, issued Dec.18, 2012, pending; which is a continuation of U.S. Pat. No. 8,165,812,issued Apr. 24, 2012, the priority dates of which are claimed and thedisclosures of which are incorporated by reference.

This invention was made with State of California support under AgreementNumber 722. The California Public Utilities Commission of the State ofCalifornia has certain rights to this invention.

FIELD

This application relates in general to power generation fleet planningand operation and, in particular, to a system and method for determiningseasonal energy consumption with the aid of a digital computer

BACKGROUND

A power grid is a geographically-distributed electricity generation,transmission, and distribution infrastructure that delivers electricityfrom power generation sources to regional and municipal power utilitiesand finally to end-consumers, including residential, commercial, andretail customers. Power generation and consumption balancing remains acrucial part of power grid and power utility planning and operations. Aselectricity is consumed almost immediately upon production, both powergeneration and power consumption must be continually balanced across theentire power grid. For instance, a power failure in one part of a powergrid could cause electrical current to reroute from remaining powergeneration sources over transmission lines of insufficient capacity andin turn create the possibility of cascading power failures andwidespread outages. As a result, the planners and operators of powergrids and power utilities need to be able to accurately gauge bothon-going and forecasted power generation from all sources, includingphotovoltaic fleets and individual photovoltaic systems, and on-goingand forecasted consumption by all consumers.

Estimating on-going and forecasted power generation requires examiningthe contribution made by each power generation system to a power grid.For instance, photovoltaic systems are widely used today forgrid-connected distributed power generation, as well as for standaloneoff-grid power systems and residential and commercial sources ofsupplemental electricity. Power grid connection of photovoltaic powergeneration is a fairly recent development. Typically, when integratedinto a power grid, photovoltaic systems are centrally operated by asupplier as a fleet, although the individual photovoltaic systems in thefleet may be deployed at different physical locations within ageographic region. Reliance on photovoltaic fleet power generation aspart of a power grid implicates the need for these photovoltaic systemsto exhibit predictable power generation behaviors, and accurate powerproduction data is needed at all levels of a power grid, including powerutilities, to which a fleet is connected.

On-going and forecasted power production data is particularly crucialwhen a photovoltaic fleet makes a significant contribution to a powergrid's overall power mix. For individual systems, power productionforecasting first involves obtaining a prediction of solar irradiance,which can be derived from ground-based measurements, satellite imagery,numerical weather prediction models, or other sources. The predictedsolar irradiance and each photovoltaic plant's system configuration iscombined with a photovoltaic simulation model to generate a forecast ofindividual photovoltaic plant power output production. The individualphotovoltaic plant forecasts can then be combined into a photovoltaicfleet forecast, such as described in commonly-assigned U.S. Pat. Nos.8,165,811; 8,165,812; 8,165,813, all issued to Hoff on Apr. 24, 2012;U.S. Pat. Nos. 8,326,535; 8,326,536, issued to Hoff on Dec. 4, 2012; andU.S. Pat. No. 8,335,649, issued to Hoff on Dec. 18, 2012, thedisclosures of which are incorporated by reference.

As photovoltaic power generation relies on solar irradiance,photovoltaic fleets operating under cloudy conditions may exhibitvariable and unpredictable performance, thereby complicating the needfor predictable power generation behaviors. Conventionally, fleetvariability is determined by collecting and feeding direct powermeasurements from individual photovoltaic systems or equivalentindirectly-derived power measurements into a centralized controlcomputer or similar arrangement. To be of optimal usefulness, the directpower measurement data must be collected in near-real time atfine-grained time intervals to enable a high resolution time series ofpower output data to be created. However, the practicality of this formof optimal approach diminishes as the number of photovoltaic systems,variations in system configurations, and geographic dispersion of thephotovoltaic fleet grow. Moreover, the costs and feasibility ofproviding remote power measurement data can make high speed datacollection and analysis insurmountable due to the bandwidth needed totransmit data and the storage space needed to contain collectedmeasurements; furthermore, the processing resources needed to scalequantitative power measurement analysis increases upwards as thephotovoltaic fleet size grows.

Similarly, for power grid and power utility planning and operationpurposes, assembling accurate photovoltaic system configurationspecification data is as important as obtaining a reliable solarirradiance forecast. The specification of a photovoltaic system'sconfiguration is typically provided by the system's owner or operator;as a result, the configuration specification data can vary in terms ofcompleteness, quality, and correctness, which can complicate or skewpower output forecasting despite the accuracy of the solar irradianceforecast. Moreover, in some situations, photovoltaic systemconfiguration specifications may simply not be available.Privately-owned residential systems, for example, are typically notcontrolled or accessible by the power grid operators and by powerutilities who need to fully understand and gauge those photovoltaicsystems' expected photovoltaic power output capabilities andshortcomings, as decreases in end-consumer operated photovoltaic systempower production could require increased power grid production tocompensate for power generation losses. Furthermore, even largeutility-connected systems may have configuration specifications that arenot publicly available due to privacy or security reasons.

To satisfy on-going and forecasted power consumption needs, the plannersand operators of power grids and power utilities require a deepunderstanding of how their customers consume and, where appropriate,generate energy. End-consumer energy consumption patterns touch uponmany of their activities, including planning for future growth,procuring power generation supply, dispatching resources, settingutility rates, and developing programs that promote power consumptionefficiency. All of these activities may also be influenced byprivately-owned photovoltaic systems that may carry part of a consumer'spower load, thereby decreasing overall expected utility electric load,or could participate as part of a power grid, thus increasing totalavailable grid power generation.

Power consumption by commercial and retail customers can be forecastwith reasonable accuracy since the kinds of activities in whichbusinesses engage in and their customary business hours are generallyknown; changes to their power consumption patterns occur with sufficientnotice, so as to enable power utilities to make any necessaryadjustments to production or resources as needed. On the other hand,accurately predicting residential power consumption remains a challenge,particularly due to location- and consumer-specific factors, such asyear-to-year weather variability, occupant comfort preferences, andbuilding internal heat gains. Residential customers, who typicallyconstitute a large part of an power utilities' consumer base, consumepower in widely varying patterns, and power consumption may be furtheraffected by on-site residential photovoltaic systems. Power utilitiesmeter total electric load only; however, any given residentialconsumer's total electric load will likely include different types ofcomponent loads, such as heating, cooling, and “always-on” baseloads,that depend on a range of disparate factors, such as the thermalperformance of their home's building envelope, occupant comfortpreferences, and internal heat gains, as well as geographic location andyear-to-year weather variability.

Analytically disaggregating total electric load into component loadtypes can provide valuable insights into residential consumer behaviorfor planning and operational purposes and can also help power utilitiesto assist customers in using energy resources efficiently to promote ahealthy environment. In particular, conventional approaches to assessingon-going and forecasted power consumption by residential power utilitycustomers generally focus on assessing building thermal load, whichtends to define a power utility's power generation system productionpeak. Building thermal loads are temperature-driven and directly relatedto building performance based on building-specific structural factors,such as HVAC (Heating, Ventilation, and Air Conditioning) type, buildinginsulation, window type, sealing quality around building penetrations,and HVAC ductwork.

Building owners typically use on-site energy audits, detailed computermodeling, or some combination of these approaches to determine theirbuilding's thermal performance. On-site energy audits require intrusivetests, such as a blower door test, and inspections and are costly andinconvenient. Building modeling requires detailed measurements, such aswindow areas, and materials inventory, such as wall insulation type.Practically, a power utility is not equipped to broadly apply thesekinds of approaches due to the coordination needed for each customerwithin an entire service territory. Alternatively, on-line energy auditsoffer residential consumers another way of determining building thermalperformance. Consumers answer a series of questions about their homesthrough on-line surveys and software analyzes their answers and makesrecommendations. Although helpful, on-line energy audits can be tediousand consumers may lose interest and abandon an audit before completion.

Therefore, a need remains for a power utility-implementable approach toremotely estimating and disaggregating on-going and expected consumerpower consumption, particularly for residential customers.

SUMMARY

A system and method to analyze building performance without requiring anon-site energy audit or customer input is described. The analysiscombines total customer energy load with externally-suppliedmeteorological data to analyze each customer's building thermalperformance, including seasonal energy consumption. Where a customeronly uses electricity supplied by a power utility, the energy load willbe based upon net (electric) load over a set time period, such asmeasured by a power utility's on-site meter for a monthly electricitybill. Where a customer also has an on-site solar power generation system(or other type of on-site power generation system) installed, the energyload will be based upon the power utility-metered net electric load plusthe solar power generated during that same time period. Note, however,that the solar power generation will need to be added to the powerutility-metered net electric load to yield the energy load in toto ininstallations where the solar power generation is not separately meteredby the power utility. Otherwise, the total customer electric load willbe underreported due to the exclusion of on-site solar power generation.The results of this analysis of each customer's building thermalperformance produces a rich dataset that includes seasonal energyconsumption. The power utility can use the dataset in planning andoperation and for other purposes.

One embodiment provides a system and method for determining seasonalenergy consumption with the aid of a digital computer. Through a powermetering energy loads for a building situated in a known location areassessed as measured over a seasonal time period. Outdoor temperaturesfor the building are assessed as measured over the seasonal time periodthrough a temperature monitoring infrastructure. A digital computercomprising a processor and a memory that is adapted to store programinstructions for execution by the processor is operated, the programinstructions capable of: expressing each energy load as a function ofthe outdoor temperature measured at the same time of the seasonal timeperiod in point-intercept form; and taking a slope of thepoint-intercept form as the fuel rate of energy consumption during theseasonal time period.

Some of the notable elements of this methodology non-exclusivelyinclude:

(1) Employing a fully derived statistical approach to generatinghigh-speed photovoltaic fleet production data;

(2) Using a small sample of input data sources as diverse asground-based weather stations, existing photovoltaic systems, or solardata calculated from satellite images;

(3) Producing results that are usable for any photovoltaic fleetconfiguration;

(4) Supporting any time resolution, even those time resolutions fasterthan the input data collection rate;

(5) Providing results in a form that is useful and usable by electricpower grid planning and operation tools;

(6) Inferring photovoltaic plant configuration specifications, which canbe used to correct, replace or, if configuration data is unavailable,stand-in for the plant's specifications;

(7) Providing more accurate operational sets of photovoltaic systemspecifications to improve photovoltaic power generation fleetforecasting;

(8) Determining whether system maintenance is required or if degradationhas occurred by comparing reported power generation to expected powergeneration;

(9) Quantifying the value of improving photovoltaic system performanceby modifying measured time series load data using estimates of a fullyperforming photovoltaic system and sending the results through a utilitybill analysis software program;

(10) Remotely determining building-specific, objective parameters usefulin modeling the building's energy consumption;

(11) Finding an “Effective R-Value” for each building, which is anintuitive comparative metric of overall building thermal performance;

(12) Assessing on-going and forecasted power consumption by a powerutility's customer base, and in enabling the power utility to adjust ormodify the generation or procurement of electric power as a function ofa power utility's remote consumer energy auditing analytical findings;

(13) Providing power utility customers with customized information toinform their energy investment decisions and identifying homes fortargeted efficiency funding; and

(14) Disaggregating total building electrical loads into individualcomponent loads, such as heating loads, cooling loads, baseloads, andother loads.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram showing a computer-implemented method forgenerating a probabilistic forecast of photovoltaic fleet powergeneration in accordance with one embodiment.

FIG. 2 is a block diagram showing a computer-implemented system forinferring operational specifications of a photovoltaic power generationsystem using net load in accordance with a further embodiment.

FIG. 3 is a graph depicting, by way of example, ten hours of time seriesirradiance data collected from a ground-based weather station with10-second resolution.

FIG. 4 is a graph depicting, by way of example, the clearness index thatcorresponds to the irradiance data presented in FIG. 3.

FIG. 5 is a graph depicting, by way of example, the change in clearnessindex that corresponds to the clearness index presented in FIG. 4.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5.

FIG. 7 is a graph depicting, by way of example, an actual probabilitydistribution for a given distance between two pairs of locations, ascalculated for a 1,000 mete×1,000 meter grid in one square meterincrements.

FIG. 8 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

FIG. 9 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions.

FIG. 10 is a graph depicting, by way of example, results by applicationof the model.

FIG. 11 is a flow diagram showing a computer-implemented method forinferring operational specifications of a photovoltaic power generationsystem in accordance with a further embodiment.

FIG. 12 is a flow diagram showing a routine 200 for simulating poweroutput of a photovoltaic power generation system 25 for use in themethod 180 of FIG. 11.

FIG. 13 is a table showing, by way of example, simulated half-hourphotovoltaic energy production for a 1-kW-AC photovoltaic system.

FIG. 14 are graphs depicting, by way of example, simulated versusmeasured power output for hypothetical photovoltaic system configurationspecifications evaluated using the method 180 of FIG. 11.

FIG. 15 is a graph depicting, by way of example, the relative meanabsolute error between the measured and simulated power output for allsystem configurations as shown in FIG. 14.

FIG. 16 are graphs depicting, by way of example, simulated versusmeasured power output for the optimal photovoltaic system configurationspecifications as shown in FIG. 14.

FIG. 17 is a flow diagram showing a computer-implemented method forinferring operational specifications of a photovoltaic power generationsystem using net load in accordance with a further embodiment.

FIG. 18 is a graph depicting, by way of example, calculation of theFuelRate and Balance Point Temperature metrics for a sample home heatedby natural gas.

FIG. 19 is a flow diagram showing a method for determining fuel ratesand balance point temperature with the aid of a digital computer inaccordance with a further embodiment.

FIG. 20 includes graphs depicting, by way of examples, average fuelconsumption versus average outdoor temperature for four configurations.

FIG. 21 is a graph depicting, by way of example, average electricityconsumption versus average outdoor temperature for the sample home inNapa, Calif. based on monthly data.

FIG. 22 is a block diagram showing, by way of examples, Solar+hometechnology blends.

FIG. 23 is a bar chart showing, by way of example, three-and-a-halfyears of consumption and production data by circuit for the sample home.

FIG. 24 is a flow diagram showing a method for performing power utilityremote consumer energy auditing with the aid of a digital computer inaccordance with a further embodiment.

FIG. 25 includes graphs showing, by way of examples, the results foreach step of the method of FIG. 24 for the sample home.

FIG. 26 includes graphs showing, by way of examples, a comparison ofresults for the sample home obtained through the Virtual Energy Auditand measured energy consumption data.

FIG. 27 is a chart showing, by way of examples, the Effective R-Valuesof the sample house before upgrades, after most upgrades, and after allupgrades.

FIG. 28 is a system for performing power utility remote consumer energyauditing with the aid of a digital computer in accordance with a furtherembodiment.

DETAILED DESCRIPTION

Overview

The planners and operators of power grids and power utilities need to beable to accurately gauge both on-going and forecasted power generationfrom all sources, including photovoltaic fleets and individualphotovoltaic systems, and on-going and forecasted consumption by allconsumers. Power generation assessment includes both obtaining areliable solar irradiance forecast and assembling accurate photovoltaicsystem configuration specification data. Power consumption assessmentincludes understanding how power utility customers consume and, whereappropriate, generate energy.

The discussion that follows is divided into three sections. The firstsection discusses photovoltaic fleet power generation forecasting, whichincludes estimating power data for a photovoltaic power generationfleet, determining point-to-point correlation of sky clearness, anddetermining point-to-point correlation of sky clearness. The secondsection discusses inferring photovoltaic power generation systemoperational specifications from net load data. The third sectiondiscusses performing power utility remote consumer energy auditing,including analyzing each customer's building thermal performance andseasonal energy consumption, and estimating and disaggregating consumerpower consumption, plus, when appropriate, the effect of on-siteresidential photovoltaic (solar) power generation on power consumptionestimates.

Photovoltaic Fleet Power Generation Forecasting

To aid with the planning and operation of photovoltaic fleets, whetherat the power grid, supplemental, or standalone power generation levels,high resolution time series of power output data is needed toefficiently estimate photovoltaic fleet power production. Thevariability of photovoltaic fleet power generation under cloudyconditions can be efficiently estimated, even in the absence of highspeed time series power production data, by applying a fully derivedstatistical approach. FIG. 1 is a flow diagram showing acomputer-implemented method 10 for generating a probabilistic forecastof photovoltaic fleet power generation in accordance with oneembodiment. The method 10 can be implemented in software and executionof the software can be performed on a computer system, such as furtherdescribed infra, as a series of process or method modules or steps.

A time series of solar irradiance or photovoltaic (“PV”) data is firstobtained (step 11) for a set of locations representative of thegeographic region within which the photovoltaic fleet is located orintended to operate, as further described infra with reference to FIG.3. Each time series contains solar irradiance observations measured orderived, then electronically recorded at a known sampling rate at fixedtime intervals, such as at half-hour intervals, over successiveobservational time periods. The solar irradiance observations caninclude solar irradiance measured by a representative set ofground-based weather stations (step 12), existing photovoltaic systems(step 13), satellite observations (step 14), or some combinationthereof. Other sources of the solar irradiance data are possible,including numeric weather prediction models.

Next, the solar irradiance data in the time series is converted overeach of the time periods, such as at half-hour intervals, into a set ofglobal horizontal irradiance clear sky indexes, which are calculatedrelative to clear sky global horizontal irradiance based on the type ofsolar irradiance data, such as described in commonly-assigned U.S. Pat.No. 10,409,925, issued Sep. 10, 2019, the disclosure of which isincorporated by reference. The set of clearness indexes are interpretedinto as irradiance statistics (step 15), as further described infra withreference to FIGS. 4-6, and power statistics, including a time series ofthe power statistics for the photovoltaic plant, are generated (step 17)as a function of the irradiance statistics and photovoltaic plantconfiguration (step 16). The photovoltaic plant configuration includespower generation and location information, including direct current(“DC”) plant and photovoltaic panel ratings; number of power inverters;latitude, longitude and elevation; sampling and recording rates; sensortype, orientation, and number; voltage at point of delivery; trackingmode (fixed, single-axis tracking, dual-axis tracking), azimuth angle,tilt angle, row-to-row spacing, tracking rotation limit, and shading orother physical obstructions. Other types of information can also beincluded as part of the photovoltaic plant configuration. The resultanthigh-speed time series plant performance data can be combined toestimate photovoltaic fleet power output and variability, such asdescribed in commonly-assigned U.S. Pat. Nos. 8,165,811; 8,165,812;8,165,813; 8,326,535; 8,335,649; and 8,326,536, cited supra, for use bypower grid planners, operators and other interested parties.

The calculated irradiance statistics are combined with the photovoltaicfleet configuration to generate the high-speed time series photovoltaicproduction data. In a further embodiment, the foregoing methodology mayalso require conversion of weather data for a region, such as data fromsatellite regions, to average point weather data. A non-optimizedapproach would be to calculate a correlation coefficient matrixon-the-fly for each satellite data point. Alternatively, a conversionfactor for performing area-to-point conversion of satellite imagery datais described in commonly-assigned U.S. Pat. Nos. 8,165,813 and8,326,536, cited supra.

Each forecast of power production data for a photovoltaic plant predictsthe expected power output over a forecast period. FIG. 2 is a blockdiagram showing a computer-implemented system 20 for generating aprobabilistic forecast of photovoltaic fleet power generation inaccordance with one embodiment. Time series power output data 32 for aphotovoltaic plant is generated using observed field conditions relatingto overhead sky clearness. Solar irradiance 23 relative to prevailingcloudy conditions 22 in a geographic region of interest is measured.Direct solar irradiance measurements can be collected by ground-basedweather stations 24. Solar irradiance measurements can also be derivedor inferred by the actual power output of existing photovoltaic systems25. Additionally, satellite observations 26 can be obtained for thegeographic region. In a further embodiment, the solar irradiance can begenerated by numerical weather prediction models. Both the direct andinferred solar irradiance measurements are considered to be sets ofpoint values that relate to a specific physical location, whereassatellite imagery data is considered to be a set of area values thatneed to be converted into point values, such as described incommonly-assigned U.S. Pat. Nos. 8,165,813 and 8,326,536, cited supra.Still other sources of solar irradiance measurements are possible.

The solar irradiance measurements are centrally collected by a computersystem 21 or equivalent computational device. The computer system 21executes the methodology described supra with reference to FIG. 1 and asfurther detailed herein to generate time series power data 26 and otheranalytics, which can be stored or provided 27 to planners, operators,and other parties for use in solar power generation 28 planning andoperations. In a further embodiment, the computer system 21 executes themethodology described infra beginning with reference to FIG. 11 forinferring operational specifications of a photovoltaic power generationsystem, which can be stored or provided 27 to planners and otherinterested parties for use in predicting individual and fleet poweroutput generation. The data feeds 29 a-c from the various sources ofsolar irradiance data need not be high speed connections; rather, thesolar irradiance measurements can be obtained at an input datacollection rate and application of the methodology described hereinprovides the generation of an output time series at any time resolution,even faster than the input time resolution. The computer system 21includes hardware components found in a general purpose programmablecomputing device, such as a central processing unit, memory, userinterfacing means, such as a keyboard, mouse, and display, input/outputports, network interface, and non-volatile storage, and execute softwareprograms structured into routines, functions, and modules for executionon the various systems. In addition, other configurations ofcomputational resources, whether provided as a dedicated system orarranged in client-server or peer-to-peer topologies, and includingunitary or distributed processing, communications, storage, and userinterfacing, are possible.

The detailed steps performed as part of the methodology described suprawith reference to FIG. 1 will now be described.

Obtain Time Series Irradiance Data

The first step is to obtain time series irradiance data fromrepresentative locations. This data can be obtained from ground-basedweather stations, existing photovoltaic systems, a satellite network, orsome combination sources, as well as from other sources. The solarirradiance data is collected from several sample locations across thegeographic region that encompasses the photovoltaic fleet.

Direct irradiance data can be obtained by collecting weather data fromground-based monitoring systems. FIG. 3 is a graph depicting, by way ofexample, ten hours of time series irradiance data collected from aground-based weather station with 10-second resolution, that is, thetime interval equals ten seconds. In the graph, the blue line 32 is themeasured horizontal irradiance and the black line 31 is the calculatedclear sky horizontal irradiance for the location of the weather station.

Irradiance data can also be inferred from select photovoltaic systemsusing their electrical power output measurements. A performance modelfor each photovoltaic system is first identified, and the input solarirradiance corresponding to the power output is determined.

Finally, satellite-based irradiance data can also be used. As satelliteimagery data is pixel-based, the data for the geographic region isprovided as a set of pixels, which span across the region andencompassing the photovoltaic fleet.

Calculate Irradiance Statistics

The time series irradiance data for each location is then converted intotime series clearness index data, which is then used to calculateirradiance statistics, as described infra.

Clearness Index (Kt)

The clearness index (Kt) is calculated for each observation in the dataset. In the case of an irradiance data set, the clearness index isdetermined by dividing the measured global horizontal irradiance by theclear sky global horizontal irradiance, may be obtained from any of avariety of analytical methods. FIG. 4 is a graph depicting, by way ofexample, the clearness index that corresponds to the irradiance datapresented in FIG. 3. Calculation of the clearness index as describedherein is also generally applicable to other expressions of irradianceand cloudy conditions, including global horizontal and direct normalirradiance.

Change in Clearness Index (ΔKt)

The change in clearness index (ΔKt) over a time increment of Δt is thedifference between the clearness index starting at the beginning of atime increment t and the clearness index starting at the beginning of atime increment t, plus a time increment Δt. FIG. 5 is a graph depicting,by way of example, the change in clearness index that corresponds to theclearness index presented in FIG. 4.

Time Period

The time series data set is next divided into time periods, forinstance, from five to sixty minutes, over which statisticalcalculations are performed. The determination of time period is selecteddepending upon the end use of the power output data and the timeresolution of the input data. For example, if fleet variabilitystatistics are to be used to schedule regulation reserves on a 30-minutebasis, the time period could be selected as 30 minutes. The time periodmust be long enough to contain a sufficient number of sampleobservations, as defined by the data time interval, yet be short enoughto be usable in the application of interest. An empirical investigationmay be required to determine the optimal time period as appropriate.

Fundamental Statistics

Table 1 lists the irradiance statistics calculated from time series datafor each time period at each location in the geographic region. Notethat time period and location subscripts are not included for eachstatistic for purposes of notational simplicity.

TABLE 1 Statistic Variable Mean clearness index μ_(Kt) Varianceclearness index σ_(Kt) ² Mean clearness index change μ_(ΔKt) Varianceclearness index change σ_(ΔKt) ²

Table 2 lists sample clearness index time series data and associatedirradiance statistics over five-minute time periods. The data is basedon time series clearness index data that has a one-minute time interval.The analysis was performed over a five-minute time period. Note that theclearness index at 12:06 is only used to calculate the clearness indexchange and not to calculate the irradiance statistics.

TABLE 2 Clearness Clearness Index Index Change (Kt) (ΔKt) 12:00 50%  40% 12:01 90%  0% 12:02 90% −80% 12:03 10%  0% 12:04 10%   80% 12:0590% −40% 12:06 50% Mean (μ) 57%  0% Variance (σ²) 13%   27%

The mean clearness index change equals the first clearness index in thesucceeding time period, minus the first clearness index in the currenttime period divided by the number of time intervals in the time period.The mean clearness index change equals zero when these two values arethe same. The mean is small when there are a sufficient number of timeintervals. Furthermore, the mean is small relative to the clearnessindex change variance. To simplify the analysis, the mean clearnessindex change is assumed to equal zero for all time periods.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5 using a half-hour hour time period.Note that FIG. 6 presents the standard deviations, determined as thesquare root of the variance, rather than the variances, to present thestandard deviations in terms that are comparable to the mean.

Calculate Fleet Irradiance Statistics

Irradiance statistics were calculated in the previous section for thedata stream at each sample location in the geographic region. Themeaning of these statistics, however, depends upon the data source.Irradiance statistics calculated from a ground-based weather stationdata represent results for a specific geographical location as pointstatistics. Irradiance statistics calculated from satellite datarepresent results for a region as area statistics. For example, if asatellite pixel corresponds to a one square kilometer grid, then theresults represent the irradiance statistics across a physical area onekilometer square.

Average irradiance statistics across the photovoltaic fleet region are acritical part of the methodology described herein. This section presentsthe steps to combine the statistical results for individual locationsand calculate average irradiance statistics for the region as a whole.The steps differ depending upon whether point statistics or areastatistics are used.

Irradiance statistics derived from ground-based sources simply need tobe averaged to form the average irradiance statistics across thephotovoltaic fleet region. Irradiance statistics from satellite sourcesare first converted from irradiance statistics for an area intoirradiance statistics for an average point within the pixel. The averagepoint statistics are then averaged across all satellite pixels todetermine the average across the photovoltaic fleet region.

Mean Clearness Index (μ _(Kt) ) and Mean Change in Clearness Index (μ_(ΔKt) )

The mean clearness index should be averaged no matter what input datasource is used, whether ground, satellite, or photovoltaic systemoriginated data. If there are N locations, then the average clearnessindex across the photovoltaic fleet region is calculated as follows.

$\begin{matrix}{\mu_{\overset{\_}{Kt}} = {\sum\limits_{i = 1}^{N}\;\frac{\mu_{{Kt}_{i}}}{N}}} & (1)\end{matrix}$

The mean change in clearness index for any period is assumed to be zero.As a result, the mean change in clearness index for the region is alsozero.μ _(ΔKt=)0  (2)

Convert Area Variance to Point Variance

The following calculations are required if satellite data is used as thesource of irradiance data. Satellite observations represent valuesaveraged across the area of the pixel, rather than single pointobservations. The clearness index derived from this data (Kt^(Area)) maytherefore be considered an average of many individual pointmeasurements.

$\begin{matrix}{{Kt}^{Area} = {\sum\limits_{i = 1}^{N}\;\frac{{Kt}^{i}}{N}}} & (3)\end{matrix}$

As a result, the variance of the area clearness index based on satellitedata can be expressed as the variance of the average clearness indexesacross all locations within the satellite pixel.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{{VAR}\left\lbrack {Kt}^{Area} \right\rbrack} = {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}\;\frac{{Kt}^{i}}{N}} \right\rbrack}}} & (4)\end{matrix}$

The variance of a sum, however, equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;{{COV}\left\lbrack {{Kt}^{i},{Kt}^{j}} \right\rbrack}}}}} & (5)\end{matrix}$

Let ρ^(Kt) ^(i) ^(,Kt) ^(j) represents the correlation coefficientbetween the clearness index at location i and location j within thesatellite pixel. By definition of correlation coefficient,COV[Kt^(i),Kt^(j)]=σ_(Kt) ^(i)σ_(Kt) ^(j)ρ^(Kt) ^(i) ^(,Kt) ^(j) .Furthermore, since the objective is to determine the average pointvariance across the satellite pixel, the standard deviation at any pointwithin the satellite pixel can be assumed to be the same and equalsσ_(Kt), which means that σ_(Kt) ^(i)σ_(Kt) ^(j)=σ_(Kt) ² for alllocation pairs. As a result, COV[Kt^(i),Kt^(j)]=σ_(Kt) ²ρ^(Kt) ^(i)^(,Kt) ^(j) . Substituting this result into Equation (5) and simplify.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{\sigma_{Kt}^{2}\left( \frac{1}{N^{2}} \right)}{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{{Kt}^{i},{Kt}^{j}}}}}} & (6)\end{matrix}$

Suppose that data was available to calculate the correlation coefficientin Equation (6). The computational effort required to perform a doublesummation for many points can be quite large and computationallyresource intensive. For example, a satellite pixel representing a onesquare kilometer area contains one million square meter increments. Withone million increments, Equation (6) would require one trillioncalculations to compute.

The calculation can be simplified by conversion into a continuousprobability density function of distances between location pairs acrossthe pixel and the correlation coefficient for that given distance, asfurther described supra. Thus, the irradiance statistics for a specificsatellite pixel, that is, an area statistic, rather than a pointstatistics, can be converted into the irradiance statistics at anaverage point within that pixel by dividing by a “Area” term (A), whichcorresponds to the area of the satellite pixel. Furthermore, theprobability density function and correlation coefficient functions aregenerally assumed to be the same for all pixels within the fleet region,making the value of A constant for all pixels and reducing thecomputational burden further. Details as to how to calculate A are alsofurther described supra.

$\begin{matrix}{{\sigma_{Kt}^{2} = \frac{\sigma_{{Kt} - {Area}}^{2}}{A_{Kt}^{{Satellite}\mspace{14mu}{Pixel}}}}{{where}\text{:}}} & (7) \\{A_{Kt}^{{Satellite}\mspace{14mu}{Pixel}} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{i,j}}}}} & (8)\end{matrix}$

Likewise, the change in clearness index variance across the satelliteregion can also be converted to an average point estimate using asimilar conversion factor, A_(ΔKt) ^(Area).

$\begin{matrix}{\sigma_{\Delta\;{Kt}}^{2} = \frac{\sigma_{{\Delta\;{Kt}} - {Area}}^{2}}{A_{\Delta\;{Kt}}^{{Satellite}\mspace{14mu}{Pixel}}}} & (9)\end{matrix}$

Variance of Clearness Index

$\left( {\sigma\frac{2}{Kt}} \right)$And Variance of Change in Clearness Index

$\left( {\sigma\frac{2}{\Delta\;{Kt}}} \right)$

At this point, the point statistics (σ_(Kt) ² and σ_(ΔKt) ²) have beendetermined for each of several representative locations within the fleetregion. These values may have been obtained from either ground-basedpoint data or by converting satellite data from area into pointstatistics. If the fleet region is small, the variances calculated ateach location i can be averaged to determine the average point varianceacross the fleet region. If there are N locations, then average varianceof the clearness index across the photovoltaic fleet region iscalculated as follows.

$\begin{matrix}{{\sigma\frac{2}{Kt}} = {\sum\limits_{i = 1}^{N}\;\frac{\sigma_{{Kt}_{i}}^{2}}{N}}} & (10)\end{matrix}$

Likewise, the variance of the clearness index change is calculated asfollows.

$\begin{matrix}{{\sigma\frac{2}{\Delta\;{Kt}}} = {\sum\limits_{i = 1}^{N}\;\frac{\sigma_{\Delta\;{Kt}_{i}}^{2}}{N}}} & (11)\end{matrix}$

Calculate Fleet Power Statistics

The next step is to calculate photovoltaic fleet power statistics usingthe fleet irradiance statistics, as determined supra, and physicalphotovoltaic fleet configuration data. These fleet power statistics arederived from the irradiance statistics and have the same time period.

The critical photovoltaic fleet performance statistics that are ofinterest are the mean fleet power, the variance of the fleet power, andthe variance of the change in fleet power over the desired time period.As in the case of irradiance statistics, the mean change in fleet poweris assumed to be zero.

Photovoltaic System Power for Single System at Time t

Photovoltaic system power output (kW) is approximately linearly relatedto the AC-rating of the photovoltaic system (R in units of kW_(AC))times plane-of-array irradiance. Plane-of-array irradiance can berepresented by the clearness index over the photovoltaic system (KtPV)times the clear sky global horizontal irradiance times an orientationfactor (O), which both converts global horizontal irradiance toplane-of-array irradiance and has an embedded factor that convertsirradiance from Watts/m² to kW output/kW of rating. Thus, at a specificpoint in time (t), the power output for a single photovoltaic system (n)equals:P _(t) ^(n) =R ^(n) O _(t) ^(n) KtPV _(t) ^(n) I _(t) ^(Clear,n)  (12)

The change in power equals the difference in power at two differentpoints in time.ΔP _(t,Δt) ^(n) =R ^(n) O _(t+Δt) ^(n) KtPV _(t+Δt) ^(n) I _(t+Δt)^(Clear,n) −R ^(n) O _(t) ^(n) KtPV _(t) ^(n) I _(t) ^(Clear,n)  (13)

The rating is constant, and over a short time interval, the two clearsky plane-of-array irradiances are approximately the same (O_(t+Δt)^(n)I_(t+Δt) ^(Clear,n)≈O_(t) ^(n)I_(t) ^(Clear,n)), so that the threeterms can be factored out and the change in the clearness index remains.ΔP _(t,Δt) ^(n) ≈R ^(n) O _(t) ^(n) I _(t) ^(Clear,n) ΔKtPV _(t)^(n)  (14)

Time Series Photovoltaic Power for Single System

P^(n) is a random variable that summarizes the power for a singlephotovoltaic system n over a set of times for a given time interval andset of time periods. ΔP^(n) is a random variable that summarizes thechange in power over the same set of times.

Mean Fleet Power (μ_(P))

The mean power for the fleet of photovoltaic systems over the timeperiod equals the expected value of the sum of the power output from allof the photovoltaic systems in the fleet.

$\begin{matrix}{\mu_{P} = {E\left\lbrack {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (15)\end{matrix}$

If the time period is short and the region small, the clear skyirradiance does not change much and can be factored out of theexpectation.

$\begin{matrix}{\mu_{P} = {\mu_{I^{Clear}}{E\left\lbrack {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}{KtPV}^{n}}} \right\rbrack}}} & (16)\end{matrix}$

Again, if the time period is short and the region small, the clearnessindex can be averaged across the photovoltaic fleet region and any givenorientation factor can be assumed to be a constant within the timeperiod. The result is that:μ_(P) =R ^(Adj.Fleet)μ_(I) _(Clear) μ _(Kt)   (17)where μ_(I) _(Clear) is calculated, μ _(Kt) is taken from Equation (1)and:

$\begin{matrix}{R^{{Adj}.{Fleet}} = {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}}}} & (18)\end{matrix}$

This value can also be expressed as the average power during clear skyconditions times the average clearness index across the region.μ_(P)=μ_(P) _(Clear) μ _(Kt)   (19)

Variance of Fleet Power (σ_(P) ²)

The variance of the power from the photovoltaic fleet equals:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (20)\end{matrix}$

If the clear sky irradiance is the same for all systems, which will bethe case when the region is small and the time period is short, then:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {I^{Clear}{\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}{KtPV}^{n}}}} \right\rbrack}} & (21)\end{matrix}$

The variance of a product of two independent random variables X, Y, thatis, VAR[XY]) equals E[X]²VAR[Y]+E[Y]²VAR[X]+VAR[X]VAR[Y]. If the Xrandom variable has a large mean and small variance relative to theother terms, then VAR[XY]≈E[X]²VAR[Y]. Thus, the clear sky irradiancecan be factored out of Equation (21) and can be written as:

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}{{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}\;{R^{n}{KtPV}^{n}O^{n}}} \right\rbrack}}} & (22)\end{matrix}$

The variance of a sum equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\;{{COV}\left\lbrack \;{{R^{i}{KtPV}^{i}O^{i}},{R^{j}{KtPV}^{j}O^{j}}} \right\rbrack}}} \right)}} & (23)\end{matrix}$

In addition, over a short time period, the factor to convert from clearsky GHI to clear sky POA does not vary much and becomes a constant. Allfour variables can be factored out of the covariance equation.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\;{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right){{COV}\left\lbrack \;{{KtPV}^{i},{KtPV}^{j}} \right\rbrack}}}} \right)}} & (24)\end{matrix}$

For any i and j, COV[KtPV^(i),KtPV^(j)]=σ_(KtPV) _(i) σ_(KtPV) _(j)ρ^(Kt) ^(i) ^(,Kt) ^(j) .

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\;{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right)\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}}\rho^{{Kt}^{i},{Kt}^{j}}}}} \right)}} & (25)\end{matrix}$

As discussed supra, the variance of the satellite data required aconversion from the satellite area, that is, the area covered by apixel, to an average point within the satellite area. In the same way,assuming a uniform clearness index across the region of the photovoltaicplant, the variance of the clearness index across a region the size ofthe photovoltaic plant within the fleet also needs to be adjusted. Thesame approach that was used to adjust the satellite clearness index canbe used to adjust the photovoltaic clearness index. Thus, each varianceneeds to be adjusted to reflect the area that the i^(th) photovoltaicplant covers.

$\begin{matrix}{\sigma_{{KtPV}^{i}}^{2} = {A_{Kt}^{i}\sigma\frac{2}{Kt}}} & (26)\end{matrix}$

Substituting and then factoring the clearness index variance given theassumption that the average variance is constant across the regionyields:

$\begin{matrix}{\sigma_{P}^{2} = {\left( {R^{{Adj}.{Fleet}}\mu_{I^{Clear}}} \right)^{2}\mspace{14mu} P^{Kt}\mspace{14mu}\sigma\frac{2}{Kt}}} & (27)\end{matrix}$where the correlation matrix equals:

$\begin{matrix}{P^{Kt} = \frac{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;{\left( {R^{i}O^{i}A_{Kt}^{i}} \right)\left( {R^{j}O^{j}A_{Kt}^{j}} \right)\rho^{{Kt}^{i},{Kt}^{j}}}}}{\left( {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}}} \right)^{2}}} & (28)\end{matrix}$

R^(Adj.Fleet)μ_(I) _(Clear) in Equation (27) can be written as the powerproduced by the photovoltaic fleet under clear sky conditions, that is:

$\begin{matrix}{\sigma_{P}^{2} = {\mu_{P^{Clear}}^{2}\mspace{14mu} P^{Kt}\mspace{14mu}\sigma\frac{2}{Kt}}} & (29)\end{matrix}$

If the region is large and the clearness index mean or variances varysubstantially across the region, then the simplifications may not beable to be applied. Notwithstanding, if the simplification isinapplicable, the systems are likely located far enough away from eachother, so as to be independent. In that case, the correlationcoefficients between plants in different regions would be zero, so mostof the terms in the summation are also zero and an inter-regionalsimplification can be made. The variance and mean then become theweighted average values based on regional photovoltaic capacity andorientation.

Discussion

In Equation (28), the correlation matrix term embeds the effect ofintra-plant and inter-plant geographic diversification. The area-relatedterms (A) inside the summations reflect the intra-plant power smoothingthat takes place in a large plant and may be calculated using thesimplified relationship, as further discussed supra. These terms arethen weighted by the effective plant output at the time, that is, therating adjusted for orientation. The multiplication of these terms withthe correlation coefficients reflects the inter-plant smoothing due tothe separation of photovoltaic systems from one another.

Variance of Change in Fleet Power (σ_(ΔP) ²)

A similar approach can be used to show that the variance of the changein power equals:

$\begin{matrix}{{\sigma_{\Delta\; P}^{2} = {\mu_{P^{Clear}}^{2}\mspace{14mu} P^{\Delta\;{Kt}}\mspace{14mu}\sigma\frac{2}{\Delta\;{Kt}}}}{{where}\text{:}}} & (30) \\{P^{\Delta\;{Kt}} = \frac{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;{\left( {R^{i}O^{i}A_{\Delta\;{Kt}}^{i}} \right)\left( {R^{j}O^{j}A_{\Delta\;{Kt}}^{j}} \right)\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{j}}}}}}{\left( {\sum\limits_{n = 1}^{N}\;{R^{n}O^{n}}} \right)^{2}}} & (31)\end{matrix}$

The determination of Equations (30) and (31) becomes computationallyintensive as the network of points becomes large. For example, a networkwith 10,000 photovoltaic systems would require the computation of acorrelation coefficient matrix with 100 million calculations. Thecomputational burden can be reduced in two ways. First, many of theterms in the matrix are zero because the photovoltaic systems arelocated too far away from each other. Thus, the double summation portionof the calculation can be simplified to eliminate zero values based ondistance between locations by construction of a grid of points. Second,once the simplification has been made, rather than calculating thematrix on-the-fly for every time period, the matrix can be calculatedonce at the beginning of the analysis for a variety of cloud speedconditions, and then the analysis would simply require a lookup of theappropriate value.

Time Lag Correlation Coefficient

The next step is to adjust the photovoltaic fleet power statistics fromthe input time interval to the desired output time interval. Forexample, the time series data may have been collected and stored every60 seconds. The user of the results, however, may want to havephotovoltaic fleet power statistics at a 10-second rate. This adjustmentis made using the time lag correlation coefficient.

The time lag correlation coefficient reflects the relationship betweenfleet power and that same fleet power starting one time interval (Δt)later. Specifically, the time lag correlation coefficient is defined asfollows:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = \frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}} & (32)\end{matrix}$

The assumption that the mean clearness index change equals zero impliesthat σ_(P) _(Δt) ²=σ_(P) ². Given a non-zero variance of power, thisassumption can also be used to show that

$\frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sigma_{P}^{2}} = {1 - {\frac{\sigma_{\Delta\; P}^{2}}{2\sigma_{P}^{2}}.}}$Therefore:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{\sigma_{\Delta\; P}^{2}}{2\sigma_{P}^{2}}}} & (33)\end{matrix}$

This relationship illustrates how the time lag correlation coefficientfor the time interval associated with the data collection rate iscompletely defined in terms of fleet power statistics alreadycalculated. A more detailed derivation is described infra.

Equation (33) can be stated completely in terms of the photovoltaicfleet configuration and the fleet region clearness index statistics bysubstituting Equations (29) and (30). Specifically, the time lagcorrelation coefficient can be stated entirely in terms of photovoltaicfleet configuration, the variance of the clearness index, and thevariance of the change in the clearness index associated with the timeincrement of the input data.

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{P^{\Delta\;{Kt}}\sigma\frac{2}{\Delta\;{Kt}}}{2\mspace{14mu} P^{Kt}\overset{\_}{\sigma\frac{2}{Kt}}}}} & (34)\end{matrix}$

Generate High-Speed Time Series Photovoltaic Fleet Power

The final step is to generate high-speed time series photovoltaic fleetpower data based on irradiance statistics, photovoltaic fleetconfiguration, and the time lag correlation coefficient. This step is toconstruct time series photovoltaic fleet production from statisticalmeasures over the desired time period, for instance, at half-hour outputintervals.

A joint probability distribution function is required for this step. Thebivariate probability density function of two unit normal randomvariables (X and Y) with a correlation coefficient of ρ equals:

$\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{2\pi\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {x^{2} + y^{2} - {2\rho\;{xy}}} \right)}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (35)\end{matrix}$

The single variable probability density function for a unit normalrandom variable X alone is

$(x) = {\frac{1}{\sqrt{2\pi}}{{\exp\left( {- \frac{x^{2}}{2}} \right)}.}}$In addition, a conditional distribution for y can be calculated based ona known x by dividing the bivariate probability density function by thesingle variable probability density, that is,

${f\left( {y❘x} \right)} = {\frac{f\left( {x,y} \right)}{f(x)}.}$Making the appropriate substitutions, the result is that the conditionaldistribution of y based on a known x equals:

$\begin{matrix}{{f\left( {y❘x} \right)} = {\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {y - {\rho\; x}} \right)^{2}}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (36)\end{matrix}$

Define a random variable

$Z = \frac{Y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}$and substitute into Equation (36). The result is that the conditionalprobability of z given a known x equals:

$\begin{matrix}{{f\left( {z❘x} \right)} = {\frac{1}{\sqrt{2\pi}}{\exp\left( {- \frac{z^{2}}{2}} \right)}}} & (37)\end{matrix}$

The cumulative distribution function for Z can be denoted by Φ(z*),where z* represents a specific value for z. The result equals aprobability (p) that ranges between 0 (when z*=−∞) and 1 (when z*=∞).The function represents the cumulative probability that any value of zis less than z*, as determined by a computer program or value lookup.

$\begin{matrix}{p = {{\Phi\left( z^{*} \right)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{z^{*}}{{\exp\left( {- \frac{z^{2}}{2}} \right)}{dz}}}}}} & (38)\end{matrix}$

Rather than selecting z*, however, a probability p falling between 0 and1 can be selected and the corresponding z* that results in thisprobability found, which can be accomplished by taking the inverse ofthe cumulative distribution function.Φ⁻¹(p)=z*  (39)

Substituting back for z as defined above results in:

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}} & (40)\end{matrix}$

Now, let the random variables equal

${X = {{\frac{P - \mu_{P}}{\sigma_{P}}\mspace{14mu}{and}\mspace{14mu} Y} = \frac{P^{\Delta\; t} - \mu_{P^{\Delta\; t}}}{\sigma_{P^{\Delta\; t}}}}},$with the correlation coefficient being the time lag correlationcoefficient between P and P^(Δt), that is, let ρ=ρ^(P,P) ^(Δt) . When Δtis small, then the mean and standard deviations for P^(Δt) areapproximately equal to the mean and standard deviation for P. Thus, Ycan be restated as

$Y \approx {\frac{P^{\Delta\; t} - \mu_{P}}{\sigma_{P}}.}$Add a time subscript to all of the relevant data to represent a specificpoint in time and substitute x, y, and ρ into Equation (40).

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{\left( \frac{P_{t}^{\Delta\; t} - \mu_{P}}{\sigma_{P}} \right) - {\rho^{P,P^{\Delta\; t}}\left( \frac{P_{t} - \mu_{P}}{\sigma_{P}} \right)}}{\sqrt{1 - \rho^{P,P^{\Delta\; t^{2}}}}}} & (41)\end{matrix}$

The random variable P^(Δt), however, is simply the random variable Pshifted in time by a time interval of Δt. As a result, at any given timet, P^(Δt) _(t)=P_(t+Δt). Make this substitution into Equation (41) andsolve in terms of P_(t+Δt).P _(t+Δt)=ρ^(P,P) ^(Δt) P _(t)+(1−ρ^(P,P) ^(Δt) )μ_(P)+√{square rootover (σ_(P) ²(1−ρ^(P,P) ^(Δt) ²))}Φ⁻¹(p)  (42)

At any given time, photovoltaic fleet power equals photovoltaic fleetpower under clear sky conditions times the average regional clearnessindex, that is, P_(t)=P_(t) ^(Clear)Kt_(t). In addition, over a shorttime period, μ_(P)≈P_(t) ^(Clear)μ _(Kt) and

$\sigma_{P}^{2} \approx {\left( P_{t}^{Clear} \right)^{2}\mspace{14mu} P^{Kt}\mspace{14mu}\sigma{\frac{2}{Kt}.}}$Substitute these three relationships into Equation (42) and factor outphotovoltaic fleet power under clear sky conditions (P_(t) ^(Clear)) ascommon to all three terms.

$\begin{matrix}{P_{t + {\Delta\; t}} = {P_{t}^{Clear}{\quad\left\lbrack {{\rho^{P,P^{\Delta\; t}}{Kt}_{t}} + {\left( {1 - \rho^{P,P^{\Delta\; t}}} \right)\mu_{\overset{\_}{Kt}}} + {\sqrt{P^{Kt}\sigma\frac{2}{Kt}\left( {1 - \rho^{P,P^{\Delta\; t^{2}}}} \right)}{\Phi^{- 1}\left( p_{t} \right)}}} \right\rbrack}}} & (43)\end{matrix}$

Equation (43) provides an iterative method to generate high-speed timeseries photovoltaic production data for a fleet of photovoltaic systems.At each time step (t+Δt), the power delivered by the fleet ofphotovoltaic systems (P_(t+Δt)) is calculated using input values fromtime step t. Thus, a time series of power outputs can be created. Theinputs include:

-   -   P_(t) ^(Clear)—photovoltaic fleet power during clear sky        conditions calculated using a photovoltaic simulation program        and clear sky irradiance.    -   Kt_(t)—average regional clearness index inferred based on P_(t)        calculated in time step t, that is, Kt_(t)=P_(t)/P_(t) ^(Clear).    -   μ _(Kt) —mean clearness index calculated using time series        irradiance data and Equation (1).

$\sigma\frac{2}{Kt}$

-   -   —variance of the clearness index calculated using time series        irradiance data and Equation (10).    -   ρ^(P,P) ^(Δt) —fleet configuration as reflected in the time lag        correlation coefficient calculated using Equation (34). In turn,        Equation (34), relies upon correlation coefficients from        Equations (28) and (31). A method to obtain these correlation        coefficients by empirical means is described in        commonly-assigned U.S. Pat. No. 8,165,811, issued Apr. 24, 2012,        and U.S. Pat. No. 8,165,813, issued Apr. 24, 2012, the        disclosure of which are incorporated by reference.    -   P^(Kt)—fleet configuration as reflected in the clearness index        correlation coefficient matrix calculated using Equation (28)        where, again, the correlation coefficients may be obtained using        the empirical results as further described infra.    -   Φ⁻¹(p_(t))—the inverse cumulative normal distribution function        based on a random variable between 0 and 1.

Discussion

The point-to-point correlation coefficients calculated using theempirical forms described supra refer to the locations of specificphotovoltaic power production sites. Importantly, note that the dataused to calculate these coefficients was not obtained from time sequencemeasurements taken at the points themselves. Rather, the coefficientswere calculated from fleet-level data (cloud speed), fixed fleet data(distances between points), and user-specified data (time interval).

The empirical relationships of the foregoing types of empiricalrelationships may be used to rapidly compute the coefficients that arethen used in the fundamental mathematical relationships. The methodologydoes not require that these specific empirical models be used andimproved models will become available in the future with additional dataand analysis.

Probability Density Function

The conversion from area statistics to point statistics relied upon twoterms A_(Kt) and A_(ΔKt) to calculate σ_(Kt) ² and σ_(ΔKt) ²,respectively. This section considers these terms in more detail. Forsimplicity, the methodology supra applies to both Kt and ΔKt, so thisnotation is dropped. Understand that the correlation coefficient ρ^(i,j)could refer to either the correlation coefficient for clearness index orthe correlation coefficient for the change in clearness index, dependingupon context. Thus, the problem at hand is to evaluate the followingrelationship:

$\begin{matrix}{A = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{i,j}}}}} & (44)\end{matrix}$

The computational effort required to calculate the correlationcoefficient matrix can be substantial. For example, suppose that the onewants to evaluate variance of the sum of points within a 1 squarekilometer satellite region by breaking the region into one millionsquare meters (1,000 meters by 1,000 meters). The complete calculationof this matrix requires the examination of 1 trillion (10¹²) locationpair combinations.

Discrete Formulation

The calculation can be simplified using the observation that many of theterms in the correlation coefficient matrix are identical. For example,the covariance between any of the one million points and themselvesis 1. This observation can be used to show that, in the case of arectangular region that has dimension of H by W points (total of N) andthe capacity is equal distributed across all parts of the region that:

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right){\quad{{\left\lbrack {{\sum\limits_{i = 0}^{H - 1}\;{\sum\limits_{j = 0}^{i}\;{{2^{k}\left\lbrack {\left( {H - i} \right)\left( {W - j} \right)} \right\rbrack}\rho^{d}}}} + {\sum\limits_{i = 0}^{W - 1}\;{\sum\limits_{j = 0}^{i}\;{{2^{k}\left\lbrack {\left( {W - i} \right)\left( {H - j} \right)} \right\rbrack}\rho^{d}}}}} \right\rbrack\mspace{76mu}{where}\text{:}\mspace{76mu} k} = \begin{matrix}{{- 1},{{{when}\mspace{14mu} i} = {{0\mspace{14mu}{and}\mspace{14mu} j} = 0}}} \\{1,{{{when}\mspace{14mu} j} = {{0\mspace{14mu}{or}\mspace{14mu} j} = i}}} \\{{2,{{{when}\mspace{14mu} 0} < j < i}}\mspace{56mu}}\end{matrix}}}}} & (45)\end{matrix}$

When the region is a square, a further simplification can be made.

$\begin{matrix}{{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}\;{\sum\limits_{j = 0}^{i}\;{2^{k}\left( {\sqrt{N} - i} \right)\left( {\sqrt{N} - j} \right)\rho^{d}}}} \right\rbrack}}\mspace{76mu}{{where}\text{:}}\mspace{79mu}{{k = \begin{matrix}{0,{{{when}\mspace{14mu} i} = {{0\mspace{14mu}{and}\mspace{14mu} j} = 0}}} \\{{2,{{{when}\mspace{14mu} j} = {{0\mspace{14mu}{or}\mspace{14mu} j} = i}}}\mspace{14mu}} \\{{3,{{{when}\mspace{14mu} 0} < j < i}}\mspace{65mu}}\end{matrix}},{and}}\mspace{76mu}{d = {\left( \sqrt{i^{2} + j^{2}} \right){\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right).}}}} & (46)\end{matrix}$The benefit of Equation (46) is that there are

$\frac{N - \sqrt{N}}{2}$rather than N² unique comminations that need to be evaluated. In theexample above, rather than requiring one trillion possible combinations,the calculation is reduced to one-half million possible combinations.

Continuous Formulation

Even given this simplification, however, the problem is stillcomputationally daunting, especially if the computation needs to beperformed repeatedly in the time series. Therefore, the problem can berestated as a continuous formulation in which case a proposedcorrelation function may be used to simplify the calculation. The onlyvariable that changes in the correlation coefficient between any of thelocation pairs is the distance between the two locations; all othervariables are the same for a given calculation. As a result, Equation(46) can be interpreted as the combination of two factors: theprobability density function for a given distance occurring and thecorrelation coefficient at the specific distance.

Consider the probability density function. The actual probability of agiven distance between two pairs occurring was calculated for a 1,000meter×1,000 meter grid in one square meter increments. The evaluation ofone trillion location pair combination possibilities was evaluated usingEquation (44) and by eliminating the correlation coefficient from theequation. FIG. 7 is a graph depicting, by way of example, an actualprobability distribution for a given distance between two pairs oflocations, as calculated for a 1,000 meter×1,000 meter grid in onesquare meter increments.

The probability distribution suggests that a continuous approach can betaken, where the goal is to find the probability density function basedon the distance, such that the integral of the probability densityfunction times the correlation coefficient function equals:A=∫f(D)ρ(d)dD  (47)

An analysis of the shape of the curve shown in FIG. 7 suggests that thedistribution can be approximated through the use of two probabilitydensity functions. The first probability density function is a quadraticfunction that is valid between 0 and √{square root over (Area)}.

$\begin{matrix}{f_{Quad} = \left\{ \begin{matrix}{\left( \frac{6}{Area} \right)\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)} & {{{for}\mspace{14mu} 0} \leq D \leq \sqrt{Area}} \\0 & {{{{for}\mspace{14mu} D} > \sqrt{Area}}\mspace{40mu}}\end{matrix} \right.} & (48)\end{matrix}$

This function is a probability density function because integratingbetween 0 and √{square root over (Area)} equals 1, that is,P[0≤D≤√{square root over (Area)}]=∫₀^(√{square root over (Area)})f_(Quad.)dD=1.

The second function is a normal distribution with a mean of √{squareroot over (Area)} and standard deviation of 0.1√{square root over(Area)}.

$\begin{matrix}{f_{Norm} = {\left( \frac{1}{0.1*\sqrt{Area}} \right)\left( \frac{1}{\sqrt{2\pi}} \right)e^{{- {(\frac{1}{2})}}{(\frac{D - \sqrt{Area}}{0.1*\sqrt{Area}})}^{2}}}} & (49)\end{matrix}$

Likewise, integrating across all values equals 1.

To construct the desired probability density function, take, forinstance, 94 percent of the quadratic density function plus 6 of thenormal density function.f=0.94∫₀ ^(√{square root over (Area)}) f _(Quad) dD+0.06∫_(−∞) ^(+∞) f_(Norm) dD  (50)

FIG. 8 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

The result is that the correlation matrix of a square area with uniformpoint distribution as N gets large can be expressed as follows, firstdropping the subscript on the variance since this equation will work forboth Kt and ΔKt.A≈[0.94∫₀ ^(√{square root over (Area)}) f _(Quad)ρ(D)dD+0.061∫_(−∞)^(+∞) f _(Norm)ρ(D)dD]  (50)where ρ(D) is a function that expresses the correlation coefficient as afunction of distance (D).

Area to Point Conversion Using Exponential Correlation Coefficient

Equation (51) simplifies the problem of calculating the correlationcoefficient and can be implemented numerically once the correlationcoefficient function is known. This section demonstrates how a closedform solution can be provided, if the functional form of the correlationcoefficient function is exponential.

An exponentially decaying function can be taken as a suitable form forthe correlation coefficient function. Assume that the functional form ofcorrelation coefficient function equals:

$\begin{matrix}{{\rho(D)} = e^{\frac{xD}{\sqrt{Area}}}} & (52)\end{matrix}$Let Quad be the solution to ∫₀^(√{square root over (Area)})f_(Quad.)ρ(D) dD.

$\begin{matrix}{{Quad} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho(D)}}} = {\left( \frac{6}{Area} \right){\int_{0}^{\sqrt{Area}}{{\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)\left\lbrack e^{\frac{xD}{\sqrt{Area}}} \right\rbrack}{dD}}}}}} & (53)\end{matrix}$Integrate to solve.

$\begin{matrix}{{Quad} = {(6)\left\lbrack {{\left( {{\frac{x}{\sqrt{Area}}D} - 1} \right)e^{\frac{xD}{\sqrt{Area}}}} - {\left( {{\left( \frac{x}{\sqrt{Area}} \right)^{2}D^{2}} - {2\frac{x}{\sqrt{Area}}D} + 2} \right)e^{\frac{xD}{\sqrt{Area}}}}} \right\rbrack}} & (54)\end{matrix}$Complete the result by evaluating at D equal to √{square root over(Area)} for the upper bound and 0 for the lower bound. The result is:

$\begin{matrix}{{Quad} = {\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} & (55)\end{matrix}$Next, consider the solution to ∫_(−∞) ^(+∞)f_(Norm.)ρ(D) dD which willbe called Norm.

$\begin{matrix}{{Norm} = {\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}{(\frac{D - \mu}{\sigma})}^{2}}e^{\frac{xD}{\sqrt{Area}}}{dD}}}}} & (56)\end{matrix}$where μ=√{square root over (Area)} and σ=0.1√{square root over (Area)}.Simplifying:

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{- {{(\frac{1}{2})}\lbrack\frac{D - {({\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}})}}{\sigma}\rbrack}^{2}}{dD}}}}} & (57)\end{matrix}$Substitute

$z = \frac{D - \left( {\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}} \right)}{\sigma}$and σdz=dD.

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}z^{2}}{dz}}}}} & (58)\end{matrix}$Integrate and solve.

$\begin{matrix}{{Norm} = e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}}} & (59)\end{matrix}$Substitute the mean of √{square root over (Area)} and the standarddeviation of 0.1√{square root over (Area)} into Equation (51).Norm=e ^(x(1+0.005x))  (60)

Substitute the solutions for Quad and Norm back into Equation (51). Theresult is the ratio of the area variance to the average point variance.This ratio was referred to as A (with the appropriate subscripts andsuperscripts) supra.

$\begin{matrix}{A = {{0.94{\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} + {0.06e^{x{({1 + {0.005x}})}}}}} & (61)\end{matrix}$

Time Lag Correlation Coefficient

This section presents an alternative approach to deriving the time lagcorrelation coefficient. The variance of the sum of the change in theclearness index equals:σ_(ΣΔKt) ²=VAR[Σ(Kt ^(Δt) −Kt)]  (62)where the summation is over N locations. This value and thecorresponding subscripts have been excluded for purposes of notationalsimplicity.Divide the summation into two parts and add several constants to theequation:

$\begin{matrix}{\sigma_{\Sigma\mspace{14mu}\Delta\;{Kt}}^{2} = {{VAR}\left\lbrack {{\sigma_{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}\left( \frac{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}{\sigma_{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}} \right)} - {\sigma_{\Sigma^{Kt}}\left( \frac{\Sigma\mspace{14mu}{Kt}}{\sigma_{\Sigma\mspace{14mu}{Kt}}} \right)}} \right\rbrack}} & (63)\end{matrix}$Since σ_(ΣKt) _(Δt) ≈σ_(ΣKt) (or σ_(ΣKt) _(Δt) if the first term in Ktand the last term in Kt^(Δt) are the same):

$\begin{matrix}{\sigma_{\Sigma\mspace{14mu}\Delta\;{Kt}}^{2} = {\sigma_{\Sigma\mspace{14mu}{Kt}}^{2}{{VAR}\left\lbrack {\frac{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}{\sigma_{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}} - \frac{\Sigma\mspace{14mu}{Kt}}{\sigma_{\Sigma\mspace{14mu}{Kt}}}} \right\rbrack}}} & (64)\end{matrix}$The variance term can be expanded as follows:

$\begin{matrix}{\sigma_{\Sigma\mspace{14mu}\Delta\;{Kt}}^{2} = {\sigma_{\Sigma\mspace{20mu}{Kt}}^{2}\left\{ {\frac{{VAR}\left\lbrack {\Sigma\mspace{14mu}{Kt}^{\Delta\; t}} \right\rbrack}{\sigma_{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}^{2}} + \frac{{VAR}\left\lbrack {\Sigma\mspace{14mu}{Kt}} \right\rbrack}{\sigma_{\Sigma\mspace{14mu}{Kt}}^{2}} - \frac{2{{COV}\left\lbrack {{\Sigma\mspace{14mu}{Kt}},{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}} \right\rbrack}}{\sigma_{\Sigma\mspace{14mu}{Kt}}\sigma_{\Sigma\mspace{14mu}{Kt}^{\Delta t}}}} \right\}}} & (65)\end{matrix}$Since COV[ΣKt,ΣKt^(Δt)]=σ_(ΣKt)σ_(ΣKt) _(Δt) ρ^(ΣKt,ΣKt) ^(Δt) , thefirst two terms equal one and the covariance term is replaced by thecorrelation coefficient.σ_(ΣΔKt) ²=2σ_(ΣKt) ²(1−ρ^(ΣKt,ΣKt) ^(Δt) )  (66)This expression rearranges to:

$\begin{matrix}{\rho^{{\Sigma\mspace{14mu}{Kt}},{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}} = {1 - {\frac{1}{2}\frac{\sigma_{\Sigma\mspace{14mu}\Delta\;{Kt}}^{2}}{\sigma_{\Sigma\mspace{14mu}{Kt}}^{2}}}}} & (67)\end{matrix}$

Assume that all photovoltaic plant ratings, orientations, and areaadjustments equal to one, calculate statistics for the clearness aloneusing the equations described supra and then substitute. The result is:

$\begin{matrix}{\rho^{{\Sigma\mspace{14mu}{Kt}},{\Sigma\mspace{14mu}{Kt}^{\Delta\; t}}} = {1 - \frac{P^{\Delta\;{Kt}}\sigma\frac{2}{\Delta\;{Kt}}}{2P^{Kt}\sigma\frac{2}{Kt}}}} & (68)\end{matrix}$

Relationship Between Time Lag Correlation Coefficient and Power/Changein Power Correlation Coefficient

This section derives the relationship between the time lag correlationcoefficient and the correlation between the series and the change in theseries for a single location.

$\mspace{76mu}{\rho^{P,{\Delta\; P}} = {\frac{{COV}\left\lbrack {P,{\Delta\; P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = {\frac{{COV}\left\lbrack {P,{P^{\Delta\; t} - P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = \frac{{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}}}}}$${{{Since}\mspace{14mu}\sigma_{\Delta\; P}^{2}} = {{{VAR}\left\lbrack {P^{\Delta\; t} - P} \right\rbrack} = {{\sigma_{P}^{2} + \sigma_{P^{\Delta\; t}}^{2} - {2{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}\mspace{14mu}{and}\mspace{14mu}{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}}} = {\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}}}},{{{then}\mspace{14mu}\rho^{P,{\Delta\; P}}} = {\frac{{\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\left( {\sigma_{P}^{2} + \sigma_{P^{\Delta\; t}}^{2} - {2\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}} \right)}}.}}$Since σ_(P) ²≈σ_(P) _(Δt) ², this expression can be further simplified.Then, square both expression and solve for the time lag correlationcoefficient:ρ^(P,P) ^(Δt) =1−2(ρ^(P,ΔP))²

Correlation Coefficients Between Two Regions

Assume that the two regions are squares of the same size, each side withN points, that is, a matrix with dimensions of √{square root over (N)}by √{square root over (N)} points, where √{square root over (N)} is aninteger, but are separated by one or more regions. Thus:

$\begin{matrix}{{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}\;{\left( \frac{1}{N^{2}} \right)\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}\;{\sum\limits_{j = {1 - \sqrt{N}}}^{\sqrt{N} - 1}\;{{k\left( {\sqrt{N} - i} \right)}\left( {\sqrt{N} - {j}} \right)\rho^{d}}}} \right\rbrack}} & (69)\end{matrix}$where

$k = \left\{ {\begin{matrix}1 & {{{when}\mspace{14mu} i} = 0} \\2 & {{{when}\mspace{14mu} i} > 0}\end{matrix},{d = {\left( \sqrt{i^{2} + \left( {j + {M\sqrt{N}}} \right)^{2}} \right)\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right)}},} \right.$and M equals the number of regions.

FIG. 9 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions. FIG. 9suggests that the probability density function can be estimated usingthe following distribution:

$\begin{matrix}{f = \left\{ \begin{matrix}{1 - \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{{for}\mspace{14mu}{Spacing}} - \sqrt{Area}} \leq D \leq {Spacing}} \\{1 + \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{for}\mspace{14mu}{Spacing}} \leq D \leq {{Spacing} + \sqrt{Area}}} \\0 & {{all}\mspace{14mu}{else}}\end{matrix} \right.} & (70)\end{matrix}$

This function is a probability density function because the integrationover all possible values equals zero. FIG. 10 is a graph depicting, byway of example, results by application of this model.

Inferring Photovoltaic System Configuration Specifications

Accurate power output forecasting through photovoltaic power predictionmodels, such as described supra, requires equally precise solarirradiance data and photovoltaic system configuration specifications.Solar irradiance data can be obtained from ground-based measurements,satellite imagery, numerical weather prediction models, as well asthrough various reliable third party sources, such as the SolarAnywhere®database service (http://www.SolarAnywhere.com), a Web-based serviceoperated by Clean Power Research, L.L.C., Napa, Calif., that can providesatellite-derived solar irradiance data forecasted up to seven daysahead of time and archival solar irradiance data, dating back to Jan. 1,1998, at time resolutions of as fast as one minute for historical dataup to several hours forecasted and then transitioning to a one-hour timeresolution up to seven days ahead of time. The SolarAnywhere® databaseservice has 20 years of hourly outdoor temperature data for everylocation in the continental United States.

On the other hand, obtaining accurate and reliable photovoltaic plantconfiguration specifications for individual photovoltaic systems can bea challenge, particularly when the photovoltaic systems are part of ageographically dispersed power generation fleet. Accurate photovoltaicsystem configuration specifications are needed to efficiently estimateindividual photovoltaic power plant production. Part of the concernarises due to an increasing number of grid-connected photovoltaicsystems that are privately-owned residential and commercial systems,where they are neither controlled nor accessible by grid operators andpower utilities, who require precise configuration specifications forplanning and operations purposes or whether they are privately-ownedutility-scale systems for which specifications are unavailable.Moreover, in some situations, the configuration specifications may beeither incorrect, incomplete or simply not available.

Method

Photovoltaic plant configuration specifications can be accuratelyinferred through analysis of historical measurements of the photovoltaicplant's production data and measured historical irradiance data. FIG. 11is a flow diagram showing a computer-implemented method 180 forinferring operational specifications of a photovoltaic power generationsystem 25 (shown in FIG. 2) in accordance with a further embodiment.Configuration data include the plant's power rating and electricalcharacteristics, including the effect of the efficiency of the modules,wiring, inverter, and other factors; and operational features, includingtracking mode (fixed, single-axis tracking, dual-axis tracking), azimuthangle, tilt angle, row-to-row spacing, tracking rotation limit, andshading or other physical obstructions. Shading and physicalobstructions can be evaluated by specifying obstructions as part of aconfiguration. For instance, an obstruction could be initially definedat an azimuth angle between 265° and 275° with a 10° elevation (tilt)angle. Additional configurations would vary the azimuth and elevationangles by fixed amounts, thereby exercising the range of possibleobstruction scenarios. The method 180 can be implemented in software andexecution of the software can be performed on a computer system 21, suchas described supra with reference to FIG. 2, as a series of process ormethod modules or steps.

Simulation

Configuration specifications can be inferred through evaluation ofmeasured historical photovoltaic system production data and measuredhistorical resource data. First, measured historical time-seriesphotovoltaic system production data and geographical coordinates arerespectively obtained for the photovoltaic power generation system 25under evaluation (steps 181 and 182). Optionally, the production datacan be segmented into multiple time periods for calculating the system'spower rating during different times of the year (step 183). A set ofphotovoltaic plant configuration specifications is then inferred foreach of the time periods, if applicable (steps 184-194), as follows.First, based on the measured historical production data, the output of anormalized 1-kW-AC photovoltaic system is simulated for the current timeperiod for a wide range of hypothetical (or model) photovoltaic systemconfigurations (step 186), as further described infra with reference toFIG. 12.

Following simulation, each of the hypothetical photovoltaic systemconfigurations is evaluated (steps 186-191), as follows. The totalmeasured energy produced over the selected time period (excluding anytimes with erroneous measured data, which are screened out duringsimulation, as explained infra) is determined (step 187). The ratio ofthe total measured energy over the total simulated energy is calculated(step 188), which produces a simulated photovoltaic system rating.However, system power ratings other than the ratio ofmeasured-to-simulated energy could be used.

Assuming that a photovoltaic simulation model that scales linearly (ornear-linearly, that is, approximately or substantially linear, such asdescribed infra beginning with reference to Equation (12)) inphotovoltaic system rating was used, each point in the simulated timeseries of power production data is then proportionately scaled up by thesimulated photovoltaic system rating (step 189). Each of the points inthe simulated and measured time series of power production data arematched up and the error between the measured and simulated power outputis calculated (step 190) using standard statistical methodologies. Forexample, the relative mean absolute error (rMAE) can be used, such asdescribed in Hoff et al., “Modeling PV Fleet Output Variability,” SolarEnergy 86, pp. 2177-2189 (2012) and Hoff et al, “Reporting of IrradianceModeling Relative Prediction Errors,” Progress in Photovoltaics: Res.Appl. DOI: 10.1002/pip.2225 (2012) the disclosure of which isincorporated by reference. Other methodologies, including but notlimited to root mean square error, to calculate the error between themeasured and simulated data could also be used. Each hypotheticalphotovoltaic system configuration is similarly evaluated (step 191).

Variance

Once all of the configurations have been explored (steps 186-191), avariance threshold is established and the variance between the measuredand simulated power outputs of all the configurations is taken (step192) to ensure that invalid data has been excluded. The hypotheticalphotovoltaic system configuration, including, but not limited to,tracking mode (fixed, single-axis tracking, dual-axis tracking), azimuthangle, tilt angle, row-to-row spacing, tracking rotation limit, andshading configuration, that minimizes error is selected (step 193). Theselected configuration represents the inferred photovoltaic systemconfiguration specification for the photovoltaic power generation system25 under evaluation for the current time period. Each time period issimilarly evaluated (step 194). Once all of the time periods have beenexplored (steps 184-194), an inferred photovoltaic system configurationspecification will have been selected for each time period. Ideally, thesame configuration will have been selected across all of the timeperiods. However, in the event of different configurations having beenselected, the configuration with the lowest overall error (step 193) canbe picked. Alternatively, other tie-breaking configuration selectioncriteria could be applied, such as the system configurationcorresponding to the most recent set of production data. In a furtherembodiment, mismatched configurations from each of the time periods mayindicate a concern outside the scope of plant configuration evaluation.For instance, the capacity of a plant may have increased, therebyenabling the plant to generate more power that would be reflected by asimulation based on the hypothetical photovoltaic system configurationswhich were applied. (In this situation, the hypothetical photovoltaicsystem configurations would have to be modified beginning at the timeperiod corresponding to the supposed capacity increase.) Still othertie-breaking configuration selection criteria are possible.

Optimization

In addition, the range of hypothetical (or model) photovoltaic systemconfigurations used in inferring the system's “optimal” configurationdata, that is, a system configuration heuristically derived throughevaluation of different permutations of configuration parameters,including power rating, electrical characteristics, and operationalfeatures, can be used to look at the effect of changing theconfiguration in view of historical measured performance. For instance,while the hypothetical configuration that minimizes error signifies theclosest (statistical) fit between actual versus simulated powergeneration models, other hypothetical configurations may offer thepotential to improve performance through changes to the plant'soperational features, such as revising tracking mode (fixed, single-axistracking, dual-axis tracking), azimuth, tilt, row-to-row spacing,tracking rotation limit, and shading configurations. Moreover, theaccuracy or degree to which a system configuration is “optimal” can beimproved further by increasing the degree by which each of theconfiguration parameters is varied. For instance, tilt angle can bepermuted in one degree increments, rather than five degrees at a time.Still other ways of structuring or permuting the configurationparameters, as well as other uses of the hypothetical photovoltaicsystem configurations, are possible.

Tuning

Optionally, the selected photovoltaic system configuration can be tuned(step 195), as further described infra with reference to FIG. 16. Theselected and, if applicable, tuned photovoltaic system configuration isthen provided (step 196) as the inferred photovoltaic systemconfiguration specifications, which can be used to correct, replace or,if configuration data is unavailable, stand-in for the system'sspecifications.

Power Output Simulation

Photovoltaic power prediction models are typically used in forecastingpower generation, but prediction models can also be used to simulatepower output for hypothetical photovoltaic system configurations. Thesimulation results can then be evaluated against actual historicalmeasured photovoltaic production data and statistically analyzed toidentify the inferred (and most probable) photovoltaic systemconfiguration specification. FIG. 12 is a flow diagram showing a routine200 for simulating power output of a photovoltaic power generationsystem 25 for use in the method 180 of FIG. 11. Power output issimulated for a wide range of hypothetical photovoltaic systemconfigurations, which are defined to exercise the different types ofphotovoltaic system configurations possible. Each of the hypotheticalconfigurations may vary based on power rating and electricalcharacteristics, including the effect of the efficiency of the solarmodules, wiring, inverter, and related factors, and by their operationalfeatures, such as size and number of photovoltaic arrays, the use offixed or tracking arrays, whether the arrays are tilted at differentangles of elevation or are oriented along differing azimuthal angles,and the degree to which each system is covered by shade on a row-to-rowbasis or due to cloud cover or other physical obstructions. Still otherconfiguration details are possible.

Initially, historical measured irradiance data for the current timeperiod is obtained (step 201), such as described supra beginning withreference to FIG. 3. Preferably, the irradiance data includes isobtained from a solar resource data set that contains both historicaland forecasted data, which allows further comparative analysis. Each ofthe hypothetical photovoltaic system configurations are evaluated (steps202-206), as follows. Optionally, the measured irradiance data isscreened (step 203) to eliminate data where observations are invalideither due to data recording issues or photovoltaic system performanceissues power output. The production data, that is, measured poweroutput, is correspondingly updated (step 204). Finally, power output issimulated based on the current system configuration and the measuredirradiance data for the current time period (step 205), such asdescribed supra beginning with reference to Equation (12). In oneembodiment, a normalized 1-kW-AC photovoltaic system is simulated, whichfacilitates proportionately scaling the simulated power output based onthe ratio (or function) of measured-to-simulated energy. A differentapproach may be required for photovoltaic simulation models that do notscale linearly (or near-linearly) with system rating. For instance, anon-linear (or non-near-linear) simulation model may need to be runmultiple times until the system rating for the particular systemconfiguration results in the same annual energy production as themeasured data over the same time period. Still other approaches toscaling non-linear (or non-near-linear) simulation model results toactual measured energy output are possible. Each system configuration issimilarly evaluated (step 206), after which power production simulationfor the current time period is complete.

Example of Inferred Photovoltaic Plant Configuration Specifications

The derivation of a simulated photovoltaic system configuration can beillustrated with a simple example. FIG. 13 is a table showing, by way ofexample, simulated half-hour photovoltaic energy production for a1-kW-AC photovoltaic system. Each column represents a differenthypothetical photovoltaic system configuration. For instance, the firstcolumn represents a horizontal photovoltaic plant with a fixed array ofsolar panels set at a 180 degree azimuth with zero tilt. Each rowrepresents the power produced at each half-hour period for a 1-kW-ACphotovoltaic system, beginning on Jan. 1, 2012 (night time half-hourperiods, when solar power production is zero, are omitted for clarity).The simulated energy production data covers the time period from Jan. 1,2012 through Dec. 31, 2012, although only the first few hours of Jan. 1,2012 are presented in FIG. 13. The latitude and longitude of thephotovoltaic system were obtained and the Solar Anywhere service, citedsupra, was used to obtain both historical and forecasted solar data andto simulate photovoltaic power output generation.

The simulated energy production can be compared to actual historicaldata. Here, in 2012, the photovoltaic plant produced 12,901,000 kWh(kilowatt hours) in total measured energy, while the hypotheticalphotovoltaic system configuration represented in the first column had asimulated output of 1,960 kWh over the same time period (for a 1-kW-ACphotovoltaic system). Assuming that a linearly-scalable (ornear-linearly scalable) photovoltaic simulation model was used, thesimulated output of 1,960 kWh implies that this particular systemconfiguration would need a rating of 6,582 kW-AC to produce the sameamount of energy, that is, 12,901,000 kWh, as the actual system. Thus,each half hour value can be multiplied by 6,582 to match simulated toactual power output.

The results can be visually presented. FIG. 14 are graphs depicting, byway of example, simulated versus measured power output 12 forhypothetical photovoltaic system configuration specifications evaluatedusing the method 180 of FIG. 11. Each of the graphs corresponds tophotovoltaic power as produced under a different hypotheticalphotovoltaic system configuration, as shown in the columns of the tableof FIG. 13. The x-axis of each graph represents measured power output inmegawatts (MW). The y-axis of each graph represents simulated poweroutput in megawatts (MW). Within each graph, the points present thehalf-hour simulated versus measured photovoltaic power data. Visually,the simulated versus measured power output data for the fixed systemconfiguration with a 180 degree azimuth angle and 15 degree tilt showsthe closest correlation.

Similarly, FIG. 15 is a graph depicting, by way of example, the rMAEbetween the measured and simulated power output for all systemconfigurations as shown in FIG. 14. The x-axis represents the percentageof rMAE for half-hour intervals. The y-axis represents the differenthypothetical photovoltaic system configurations. Again, the fixed systemconfiguration with a 180 degree azimuth angle and 15 degree tiltreflects the lowest rMAE and accordingly provides an optimal systemconfiguration.

Optimizing Photovoltaic System Configuration Specifications

Truly perfect weather data does not exist, as there will always beinaccuracies in weather data, whether the result of calibration or othererrors or incorrect model translation. In addition, photovoltaic plantperformance is ultimately unpredictable due to unforeseeable events andcustomer maintenance needs. For example, a power inverter outage is anunpredictable photovoltaic performance event, while photovoltaic panelwashing after a long period of soiling is an example of an unpredictablecustomer maintenance event.

Tuning

In a further embodiment, the power calibration model can be tuned toimprove forecasts of power output of the photovoltaic plant based on theinferred (and optimal) photovoltaic plant configuration specification,such as described in commonly-assigned U.S. Pat. No. 10,409,925, issuedSep. 10, 2019, the disclosure of which is incorporated by reference.Tuning helps to account for subtleties not incorporated into theselected photovoltaic simulation model, including any non-linear (ornon-near-linear) issues in the power model. FIG. 16 are graphsdepicting, by way of example, simulated versus measured power output forthe optimal photovoltaic system configuration specifications as shown inFIG. 14. The graphs corresponds to photovoltaic power as produced underthe optimal photovoltaic system configuration, that is, a fixed systemconfiguration with a 180 degree azimuth angle and 15 degree tilt before(left graph) and after (right graph) tuning. The x-axis of each graphrepresents measured power output in megawatts (MW). The y-axis of eachgraph represents simulated power output in megawatts (MW).

Referring first to the “before” graph, the simulated power productiondata over-predicts power output during lower measured power conditionsand under-predicts power output during high measured power conditions.Referring next to the “after” graph, tuning removes the uncertaintyprimarily due to irradiance data error and power conversioninaccuracies. As a result, the rMAE (not shown) is reduced from 11.4percent to 9.1 percent while also eliminating much of the remainingbias.

Inferring Photovoltaic System Configuration Specifications Using NetLoad Data

The historical measured photovoltaic power production data, which isnecessary for inferring operational photovoltaic system configurationspecifications per the approach described supra beginning with referenceto FIG. 11, may not always be available. High-quality, historical,measured photovoltaic production data may be unavailable for severalreasons. For example, photovoltaic system production may not be directlymonitored. Alternatively, photovoltaic system production may bemonitored by one party, such as the photovoltaic system owner, but theproduction data may be unavailable to other interested parties, such asan electric utility. Moreover, even where a photovoltaic systemproduction is monitored and production data is available, the quality ofthe production data may be questionable, unreliable, or otherwiseunusable. These situations are particularly prevalent with photovoltaicsystems located on the premises of residential utility customers.

In a further embodiment, net energy (electric) load data for a buildingcan be used to infer operational photovoltaic system configurationspecifications, as an alternative to historical measured photovoltaicsystem production data. Net load data can also be used, in combinationwith measured outdoor temperature data over the same period, toperforming power utility remote consumer energy auditing, as furtherdiscussed infra beginning with reference to FIG. 19.

Smart electric meters provide one source of net load data. Smart metersare becoming increasingly commonplace, as power utilities move towardstiered and time-of-use electricity pricing structures, which requireknowledge of when and how much power is consumed based on the time ofday and season. Smart meters also allow a power utility to monitor thenet power load of a building, but generally not the component powerloads of individual appliances or machinery (collectively,“components”). Note that smart meters are also available for measuringor monitoring other types of energy, fuel, or commodity consumption orusage, including natural gas, liquid propane, and water.

Typically, a smart meter is interposed on the electric power line thatsupplies electricity to a building. The smart meter includes powersensing circuitry to measure power consumption within the building andrecordation circuitry to cumulatively or instantaneously record thepower consumption as net load data. In addition, the smart meterincorporates a bidirectional communications interface with which toconnect to and interoperate with a server or other computationalinfrastructure operated by or on behalf of the power utility. Thecommunications interface can be through wired or wireless means. Thecommunications interface also offers the ability for the utility toremotely reduce load, disconnect-reconnect service, and interface toother utility service meters, such as gas and water meters. The smartmeter could also be accessible by other systems, where permitted by thepower utility, such as the resident of the building or third partiesinterested in obtaining or monitoring power consumption.

Net Load Characterization

Smart meters typically record detailed time series data for individualcustomers. In most cases, the various component loads are not directlymeasurable; component load measurement would require the smart meter tobe able to identify when specific components began and ceased operation,which is largely impracticable. As a result, only net load is availableand individual component loads must be estimated.

Assuming that a building has only one point of electricity service, netelectricity load during any given time interval, such as measured by asmart meter, equals the sum of multiple component loads, minus anyon-site distributed generation. For the sake of discussion, aphotovoltaic system will be assumed to provide all on-site distributedgeneration, although other sources of on-site distributed generation arepossible. Only one photovoltaic system is necessary, because the outputfrom any individual photovoltaic systems situated in the same locationwould be correlated. Thus, the photovoltaic system performance can bespecified by a single operational photovoltaic system configuration.

Component Loads

Individual component loads represent the load associated with groups ofdevices with similar load characteristics. For example, all lights onthe same circuit are associated with a single component load because thelight work in tandem with each other.

Component loads can be characterized into three types. A Base Loadrepresents constant power that is drawn at all times. A Binary Loadrepresents a load that is either on or off, and which, when on, drawspower at a single relatively stable power level. For example, the powerdrawn by a refrigerator is a binary load. Finally, a Variable Loadrepresents a load that can take on multiple power levels. For instance,the power drawn by an electric range is variable, as the load depends onthe number of stove burners in use and their settings.

In any given building, there is one base load, one or more binary loads,and one or more variable loads. The base load equals the sum of allcomponent loads that are on at all times. The binary loads equal the sumof the all component loads that are binary during the time that thecomponents are on. Each binary load can be expressed as the product ofan indicator function, that is, a value that is either 0 or 1, and abinary load level (energy consumption) for that binary load. Whenmultiplied by the binary load level, the indicator function acts as anidentify function that returns the binary load level only when the valueof the indicator function is 1. When the value of the indicator functionis 0, the binary load level is 0, which masks out the binary load. Thevariable loads equal the sum of all component loads that are variableduring the time that the components are on based on their correspondingvariable load levels (energy consumption).

A Net Load at time interval t can be expressed as:

$\begin{matrix}{{{Net}\mspace{14mu}{Load}_{t}} = {{{Base}\mspace{14mu}{Load}} + {\sum\limits_{m = 1}^{M}\;{1_{t}^{m} \times {Binary}\mspace{14mu}{Load}^{m}}} + {\sum\limits_{n = 1}^{N}\;{{Variable}\mspace{14mu}{Load}_{t}^{n}}} - {PV}_{t}}} & (71)\end{matrix}$where Base Load represents the base load, M is the number of binaryloads, Binary Load^(m) represents component binary load m, 1_(t) ^(m) isan indicator function at time interval t, N is the number of variableloads, Variable Load_(t) ^(n) represents component variable load m attime interval t. There is no time subscript on Base Load because thebase load is the same at all times. The indicator function (1_(t) ^(m))is either 0 or 1 for component Binary Load^(m) at time interval t. Thevalue is 0 when the load is off, and the value is 1 when the load on.There is a time subscript (t) on the indicator function, but there is notime subscript on the Binary Load proper because the binary load isconstant when on.

Photovoltaic Production

Photovoltaic production can be solved by rearranging Equation (71) ifthe net load and all individual load components are known at a giventime interval. Photovoltaic production PV_(t) at time interval t can berepresented by the normalized photovoltaic production for a 1-kWphotovoltaic system for a particular configuration times the rating ofthe system, such that:PV _(t)=(Rating)( PV _(t))  (72)where Rating is the rating of the photovoltaic system in kilowatts (kW),and PV _(t) is the production associated with a normalized 1-kWphotovoltaic system for a particular photovoltaic system configuration.

Given an accurate photovoltaic simulation model, the normalizedphotovoltaic production PV _(t) at time interval t can be expressed as afunction of photovoltaic system configuration and solar resource data.The photovoltaic system configuration is not dependent on time, whilethe solar resource data is dependent on time:PV _(t) =f(Config,Solar_(t))  (73)where Config represents a set of photovoltaic system configurationparameters, for instance, azimuth, tilt, tracking mode, and shading, fora normalized 1-kW photovoltaic system and Solar is the solar resourceand other weather data, including normalized horizontal irradiation,average ambient temperature, and wind speed, at time interval t.

Substitute Equation (73) into Equation (72), then into Equation (71):

$\begin{matrix}{{{Net}\mspace{14mu}{Load}_{t}} = {{{Base}\mspace{14mu}{Load}} + {\sum\limits_{m = 1}^{M}\;{1_{t}^{m} \times {Binary}\mspace{14mu}{Load}^{m}}} + {\sum\limits_{n = 1}^{N}\;{{Variable}\mspace{14mu}{Load}_{t}^{n}}} - {({Rating}) \times {f\left( {{Config},{Solar}_{t}} \right)}}}} & (74)\end{matrix}$

Estimate Component Loads

In most cases, the various component loads, that is, the binary loadsand the variable loads, are not directly measurable and only the netload is available. As a result, the component loads must be estimated.

Simple Case

In the simplest scenario, there is only a base load and the net loadwill directly correspond to the base load. Adding a binary componentload complicates the simplest scenario. For purposes of illustrations,assume one binary load Binary Load* and Equation (74) simplifies to:Net Load_(t)=Base Load+1_(t)*×BinaryLoad*−(Rating)×f(Config.,Solar_(t))  (75)

Consider the analysis over a 24-hour period using a one-hour timeinterval. Solving Equation (75) yields an array of 24 net load values:

$\begin{matrix}{\begin{bmatrix}{{Net}\mspace{14mu}{Load}_{1}} \\\cdots \\{{Net}\mspace{14mu}{Load}_{24}}\end{bmatrix} = {{{Base}\mspace{14mu}{{Load}\begin{bmatrix}1 \\\cdots \\1\end{bmatrix}}} + {{Binary}\mspace{14mu}{{Load}^{*}\begin{bmatrix}1_{1}^{*} \\\cdots \\1_{24}^{*}\end{bmatrix}}} - {({Rating})\begin{bmatrix}{f\left( {{{Config}.},{Solar}_{1}} \right)} \\\cdots \\{f\left( {{{Config}.},{Solar}_{24}} \right)}\end{bmatrix}}}} & (76)\end{matrix}$where Net Load_(t) is the net load at time interval t. The system of 24equations expressed by Equation (76) has many unique variables and isdifficult to solve.

When the variables on the right-hand side of Equation (76) areparameterized into a set of key parameters, Equation (76) can be used toestimate net load for each time period. The key parameters include theBase Load, any Binary Loads, an Variable Loads (not shown in Equation(76)), photovoltaic system configurations (Config.), and solar resourceand other weather data (Solar). The photovoltaic system ratings includepower ratings hypothesized for the plant for which a net load is beingestimated. Other key parameters are possible.

Simplifying Assumptions

This solution space of Equation (76) can be reduced in several ways.First, the photovoltaic production values are not 24 unrelated hourlyvalues. Rather, the values are related based on photovoltaic systemconfiguration and weather data input. Given accurate weather data, the24 photovoltaic values reduce to only one unknown variable, which issystem orientation. Second, a particularly interesting type of binaryload is a temperature-related binary load, which is related to the timeof day. The indicator function (1_(t)*) for a time-related binary loadequals 1 when the hours are between h₁ and h₂, and 0 for all otherhours. As a result, rather than requiring 24 values, only h₁ and h₂ arerequired to find the binary load.

Thus, Equation (76) can be simplified to require only two types of data,historical time series data, as expressed by net load and solar resourcedata, and a set of unknown parameters, which include photovoltaic systemrating, photovoltaic system configuration, base load, binary load,binary load indicator function start hour h₁, and binary load indicatorfunction end hour h₂.

Net loads are typically measured by a power utility on an hourly basis,although other net load measurement intervals are possible. The solarirradiance data, as well as simulation tools, can be provided by thirdparty sources, such as the SolarAnywhere data grid web interface, which,by default, reports irradiance data for a desired location using asingle observation time, and the SolarAnywhere photovoltaic systemmodeling service, available in the SolarAnywhere Toolkit, that useshourly resource data and user-defined physical system attributes tosimulate configuration-specific photovoltaic system output.SolarAnywhere is available online (http://www.SolarAnywhere.com) throughWeb-based services operated by Clean Power Research, L.L.C., Napa,Calif. Other sources of the solar irradiance data are possible,including numeric weather prediction models.

Total Squared Error

Let Net Load_(t) ^(Estimated) represent the estimated net load at timeinterval t based on the key parameters input to the right-hand side ofEquation (76), as described supra. The total squared error associatedwith the estimation equals:

$\begin{matrix}{{{Total}\mspace{14mu}{Squared}\mspace{14mu}{Error}} = {\sum\limits_{t = 1}^{24}\;\left( {{{Net}\mspace{14mu}{Load}_{t}} - {{Net}\mspace{14mu}{Load}_{t}^{Estimated}}} \right)^{2}}} & (77)\end{matrix}$The key parameters should be selected to minimize the Total SquaredError, using a minimization approach, such as described supra withreference to FIG. 11.

Method

Photovoltaic plant configuration specifications can be accuratelyinferred with net load data applied to minimize total squared error.FIG. 17 is a flow diagram showing a computer-implemented method 220 forinferring operational specifications of a photovoltaic power generationsystem 25 (shown in FIG. 2) using net load data in accordance with afurther embodiment. The method 180 can be implemented in software andexecution of the software can be performed on a computer system 21, suchas described supra with reference to FIG. 2, as a series of process ormethod modules or steps.

As a preliminary step, time series net load data is obtained (step 221),which could be supplied, for instance, by a smart meter that monitor thenet power load of a building. Other source of net load data arepossible. An appropriate time period is then selected (step 222).Preferably, a time period with minimum or consistent power consumptionis selected. Longer duration, possibly contiguous time periods providebetter results, than shorter duration, temporally-distinct time periods.For residential applications, such time periods correspond to when theoccupants are on vacation or away from home for an extended period oftime. For commercial applications, such time periods correspond to aweekend or holiday when employees are away from work. Still otherappropriate time periods are possible.

Next, based on the historical solar resource and other weather data, theoutput of a normalized 1-kW-AC photovoltaic system is simulated for thecurrent time period for a wide range of hypothetical (or model)photovoltaic system configurations (step 223), as further describedsupra with reference to FIG. 12. Power generation data is simulated fora range of hypothetical photovoltaic system configurations based on anormalized solar power simulation model. Net load data is estimatedbased on a base load and, if applicable, any binary loads and anyvariable loads net load is estimated (step 224) by selecting keyparameters, per Equation (76). The key parameters include the base load,any binary loads, any variable loads, photovoltaic systemconfigurations, and solar resource and other weather data. Thephotovoltaic system ratings include power ratings hypothesized for theplant. Other key parameters are possible. As explained supra, a specialcase exists when there is only one binary load that is bothtemperature-related and only occurring between certain hours of the day.As well, a special case exists when there are no variable loads.

Finally, total squared error between the estimated and actual net loadfor each time period is minimized (step 225). The set of key parameterscorresponding to the net load estimate that minimizes the total squarederror with the measured net load data, per Equation (77), represents theinferred specifications of the photovoltaic plant configuration.

Photovoltaic system configurations are included as one of the keyparameters. A set of hypothetical photovoltaic system configurations aredefined that include power ratings and operational features, including,but not limited to, tracking mode (fixed, single-axis tracking,dual-axis tracking), azimuth angle, tilt angle, row-to-row spacing,tracking rotation limit, and shading configuration. The selectedconfiguration represents the inferred photovoltaic system configurationspecification for the photovoltaic power generation system 25 underevaluation for the current time period. In turn, the photovoltaic systemconfiguration that is part of the set of key parameters that minimizethe total squared error will become the inferred system specification.

Estimating and Disaggregating Consumer Power Consumption

Power utilities need to have a comprehensive understanding of how andwhen their customers consume energy as disaggregated from overallcustomer energy load into individual categories of component loads(discussed in detail supra in the section entitled, “Component Loads”)for planning, operational, and other purposes. Customer energyconsumption patterns touch upon many of a power utility's activities,both long- and short-term. By performing a remote energy audit of theircustomers, preferably over a sufficiently representative sample size oftheir customer base, a power utility can improve the balancing of powergeneration, procurement, and output by their power generation equipmentand energy sources under their control against their customers' totalconsumption as categorized by disaggregated component load.

In addition, power utilities need to understand how their customersproduce and consume energy that has been generated on-site throughphotovoltaic (solar) or other power generation means. Some powerutilities have installed separate power meters to track both theelectricity that they have supplied and on-site solar (or other) powergeneration. As a result, these power utilities have the ability tocalculate each customer's energy load by combining the two observed(electricity and solar) power meter readings; however, those powerutilities that only have a traditional single-power meter installationat each customer location are unable to track their customer's on-sitepower generation, as the on-site-generated power is typically fed intothe customer's electricity circuits before their (single) power meter.Thus, the energy contribution made by on-site power generation iseffectively concealed and, without more, the customer's energy loadremains unascertainable by the power utility.

The discussion of consumer power consumption analysis is divided intothree parts. First, the thermal component of the analysis is discussedbased on established principles of heat transfer, such as the notion ofan overall heat transfer coefficient that governs the heat lossresulting from the difference between indoor and outdoor temperatures.Second, a new building performance metric, called “Effective R-Value,”is empirically determined to summarize a building's ability to resistthermal losses; the Effective R-Value is analogous to the R-value usedin building insulation materials. Third, demonstrates how to observeBalance Point Temperature empirically without knowing indoor temperatureor internal heat gains.

In this section, except as noted, “fuel” and “energy” may be usedinterchangeably, where fuel generally refers to a non-electric,physically consumable deliverable and energy generally refers toelectricity, with similar meaning applied to compound terms, such as“FuelRate,” that is, FuelRate could be based on a consumable fuel, likenatural gas, or on electricity.

Thermal Components

Characterizing building performance based on the absolute amount ofenergy consumed can be limiting. First, total energy consumption is afunction of weather variability. Cold winters result in increasedheating fuel consumption and hot summers result in increased coolingfuel consumption. Second, total energy consumption is a function of theoccupants' temperature preferences. Some occupants like their homeswarmer in the winter than other occupants, cooler in the summer, orboth. Third, total energy consumption is a function of internal heatgains. Some homes have high internal heat gains due to good solaraccess, many occupants, a substantial amount of waste heat from electricdevices operated in the home, or some combination of the foregoingfactors. These limitations make reliance on absolute energy consumptionimpracticable where a power utility desires to disaggregate intoindividual component loads.

Thermal component loads are directly related to building performance andare influenced by factors like HVAC type and ductwork, buildinginsulation, window type, and sealing quality around structurepenetrations. The analysis discussed in this section empirically andobjectively characterizes building performance using utility-meteredenergy consumption data in combination with externally-suppliedmeteorological data. The analysis uses the SolarAnywhere® databaseservice, cited supra, for meteorological data for all locations,although other sources of meteorological data are possible.

To be of use to power utilities, the results of the audit must beobjective and independent of year-to-year weather variability, occupantcomfort preferences, and internal heat gains, which is related to solaraccess, the number of occupants, and the waste heat produced byelectrical devices in the home. Specifically, the FuelRate metric hasbeen developed to meets these objective requirements. FIG. 18 is a graphdepicting, by way of example, calculation of the FuelRate and BalancePoint Temperature metrics for a sample home heated by natural gas. Thex-axis represents average outdoor temperature in degrees Fahrenheit. They-axis represents average natural gas consumption in Btu per hour(Btu/hr). The FuelRate represents the effectiveness of a home inmaintaining internal comfort levels and has units of Btu/hr-° F. TheBalance Point Temperature is a known metric that represents the outdoorair temperature when the heat gains of the building are equal to theheat losses.

In this example, natural gas is consumed during the winter months 231,but not during the summer months 232; there is minimal natural gasconsumption during the months falling in-between 233 the winter 231 and232. The average amount of natural gas consumed is plotted against theaverage outdoor temperature to form a line 230, whose slope during thewinter months 231 is the FuelRate. The Balance Point Temperature isshown at the “elbow” of line 230, where natural gas consumption iscurtailed. The home does not consume any heating fuel above the BalancePoint Temperature. Note that the FuelRate can be calculated from totalmetered energy (natural gas) load, even though the home uses natural gasfor other non-space heating purposes, such as stovetop cooking and waterheating. A similar curve can be constructed for electric cooling; thecustomer's annual heating and cooling usage could then be disaggregated,as further discussed infra.

The methodology can be used for homes heated using natural gas,electricity, or any other fuel based on hourly, daily, or monthlymeasured consumption data. The methodology also applies to the coolingseason. By way of further examples, homes located in California,Washington, and New York, incorporating heating by natural gas, electricresistance, and electric heat pump technologies, are discussed infra.The examples demonstrate a linear behavior in all these diverse casesthat is broadly applicable by location and fuel.

Instantaneous Heat Transfer

A building's thermal performance can be characterized by its overallheat transfer coefficient and its total exterior surface area(A_(Total)). The product of these two values equals the building'sthermal conductivity (UA_(Total)).

A building's total instantaneous rate of heat loss at time t ({dot over(Q)}_(t)) through the walls, ceilings, floors, and windows, equals thebuilding's thermal conductivity UA_(Total) times the difference betweenthe indoor and outdoor temperatures. Assuming that the indoortemperature is constant to obviate the need for a time subscript yields:{dot over (Q)} _(t) =UA ^(Total)(T ^(Indoor) −T _(t) ^(Outdoor))  (78)

Equation (78) expresses the relationship that the building loses heat ifthe indoor temperature exceeds the outdoor temperature, that is, {dotover (Q)}_(t) is a positive value, whereas the building gains heat ifthe outdoor temperature exceeds the indoor temperature, that is, {dotover (Q)}_(t) is a negative value.

Daily Analysis

A building has thermal mass. As a result, a building provides thermalstorage, so that changes in the indoor temperature have a lowermagnitude than changes in the outdoor temperature. For purposes ofthermal analysis, daily heat loss, which occurs over the course of aday, is more relevant rather instantaneous heat loss. Daily heat lossequals the integration of Equation (78) over one day:Q _(day) =UA ^(Total)(24)(T ^(Indoor) −T _(day) ^(Outdoor))  (79)where T _(day) ^(Outdoor) is the average daily temperature.

Balance Point Temperature Definition

A building is in thermal equilibrium when the indoor temperature isneither increasing nor decreasing. Let the Balance Point Temperature(T^(Balance Point)) correspond to the outdoor temperature at which thebuilding is in equilibrium without any auxiliary heating or cooling. TheBalance Point Temperature reflects both the indoor temperature selectedby occupants based on their comfort preferences, as well as the heatgain from internal sources, such as device usage, for instance, lights,computers, and so forth, body heat given off by the occupants, and solargain through windows and other building surfaces. The Balance PointTemperature is always less than the indoor temperature since internalgains are always positive.

The mathematical development in this section makes two assumptions forpurposes of clarity. First, only one Balance Point Temperature isassumed for the year. Most buildings will have different balance pointtemperatures for the heating and cooling seasons. The formulas describedherein can be adjusted to reflect more than one balance pointtemperature. Second, internal heat gain are assumed to be constantacross the year, which means that the Balance Point Temperature isconstant.

Components of Daily Heat Loss

Expanding Equation (79) to incorporate Balance Point Temperature yields:Q _(day) =UA ^(Total)(24)(T ^(Indoor) −T ^(Balance Point) +T^(Balance Point) −T _(day) ^(Outdoor))  (80)

Divide daily heat loss into two parts, one part that is independent ofoutdoor temperature and one part that is dependent on outdoortemperature:Q _(day) =UA ^(Total)(24)(T ^(Indoor) −T ^(Balance Point))+UA^(Total)(24)(T ^(Balance Point) −T _(day) ^(Outdoor))  (81)

The first term of Equation (81) is always positive and is constant,given the assumptions stated above. The first term is based on thedifference between the indoor temperature and the Balance PointTemperature. The second term can be positive or negative. The buildingloses heat on days when the average outdoor temperature is less than theBalance Point Temperature. The building gains heat on days when theaverage outdoor temperature exceeds the Balance Point Temperature.

Rewriting Equation (81) to have three terms that are always positiveyields:Q _(day) =UA ^(Total)(24)(T ^(Indoor) −T ^(Balance Point))+UA^(Total)(24)max(T ^(Balance Point) −T _(day) ^(Outdoor),0)−UA^(Total)(24)max( T _(day) ^(Outdoor) −T ^(Balance Point),0)  (81)

The first term of Equation (82) is the heat loss when the building is intemperature equilibrium. The second term is the additional heat losswhen average outdoor temperature is less than the Balance PointTemperature. The third term is the heat gain when average outdoortemperature exceeds the Balance Point Temperature.

Heat Loss Over a Period

Heat loss over a selected period with N days equals Equation (82) summedover N days. Summing Equation (82) over N days and rearranging theresult yields:

$\begin{matrix}{{{Q_{Period} - {(24)(N)\left( {\overset{\_}{Q}}^{{Internal}\mspace{14mu}{Gains}} \right)}} = {\overset{\overset{{Requires}\mspace{14mu}{Heating}\mspace{14mu}{Fuel}}{︷}}{\left( {UA}^{Total} \right)(24)({HDD})} - \overset{\overset{{Requires}\mspace{14mu}{Cooling}\mspace{14mu}{Fuel}}{︷}}{\left( {UA}^{Total} \right)(24)({CDD})}}}\mspace{76mu}{{where}\text{:}}} & (83) \\{\mspace{76mu}{{\overset{\_}{Q}}^{{Internal}\mspace{14mu}{Gains}} = {\left( {UA}^{Total} \right)\left( {T^{Indoor} - T^{{Balance}\mspace{14mu}{Point}}} \right)}}} & (84) \\{\mspace{76mu}{{HDD} = {\sum\limits_{{day} = 1}^{N}\;{\max\left( {{T^{{Balance}\mspace{14mu}{Point}} - {\overset{\_}{T}}_{day}^{Outdoor}},0} \right)}}}} & (85) \\{\mspace{76mu}{{CDD} = {\sum\limits_{{day} = 1}^{N}\;{\max\left( {{{\overset{\_}{T}}_{day}^{Outdoor} - T^{{Balance}\mspace{14mu}{Point}}},0} \right)}}}} & (86)\end{matrix}$

The Balance Point Temperature can be calculated by solving Equation(84). The Balance Point Temperature equals indoor temperature minusaverage hourly internal gains divided by the building's thermalconductivity. The Balance Point Temperature incorporates the effect ofboth occupant temperature preferences and internal heat gains.

$\begin{matrix}{T^{{Balance}\mspace{14mu}{Point}} = {T^{Indoor} - \left( \frac{{\overset{\_}{Q}}^{{Internal}\mspace{14mu}{Gains}}}{{UA}^{Total}} \right)}} & (87)\end{matrix}$

Heat Fuel Requirement

Consider the heating fuel term on the right-hand side of Equation (83).An HVAC system consumes fuel to deliver the required amount of heat,which equals the required heat divided by the heating system efficiency.

$\begin{matrix}{\begin{matrix}{{Fuel}^{Heating} = \frac{\left( {UA}^{Total} \right)(24)({HDD})}{\eta^{Heating}}} \\{= {\left( {FuelRate}^{Heating} \right)(24)({HDD})}}\end{matrix}{{where}\text{:}}} & (88) \\{{FuelRate}^{Heating} = \frac{{UA}^{Total}}{\eta^{Heating}}} & (89)\end{matrix}$

A similar calculation applies to the cooling fuel term on the right-handside of Equation (83). Whether for heating or cooling, the efficiencyterms include equipment efficiency and efficiency losses due to heatingor cooling distribution.

Heating Degree Days (HDD) and Cooling Degree Days (CDD)

HDDs and CDDs are typically calculated for a one-year period; however,they could also be calculated over shorter time periods, such as a day,week, or month. By selecting a time-period, p, that has D days, suchthat the average daily temperature never exceeds the balance point,Equation (85) simplifies to:HDD _(p) =D(T ^(Balance Point) −T _(p) ^(Outdoor))  (90)where T _(p) ^(Outdoor) is the average outdoor temperature over theselected period.

Substituting Equation (90) into Equation (88) for a period when thesubstitution is valid yields:Fuel_(p) ^(Heating)=(FuelRate^(Heating))(24)(D)(T ^(Balance Point) −T_(p) ^(Outdoor))  (91)Divide by 24*D and rearrange to yield the average rate of heating fuelconsumption over the selected period:

$\begin{matrix}{{AvgFuel}_{p}^{Heating} = {{{- \left( {FuelRate}^{Heating} \right)}\overset{\;}{\left( {\overset{\_}{T}}_{p}^{Outdoor} \right)}} + {\left( {FuelRate}^{Heating} \right)\left( T^{{Balance}\mspace{14mu}{Point}} \right)}}} & (92)\end{matrix}$Similarly, the average rate of cooling fuel consumption equals:

$\begin{matrix}{{{AvgFuel}_{p}^{Cooling} = {{\left( {FuelRate}^{Cooling} \right)\left( {\overset{\_}{T}}_{p}^{Outdoor} \right)} - {\left( {FuelRate}^{Cooling} \right)\left( T^{{Balance}\mspace{14mu}{Point}} \right)}}}\mspace{76mu}{{where}\text{:}}} & (93) \\{\mspace{76mu}{{FuelRate}^{Cooling} = \frac{{UA}^{Total}}{\eta^{Cooling}}}} & (94)\end{matrix}$

FuelRate^(Heating) and FuelRate^(Cooling) are two key parameters thatdefine building performance in the analysis. Equations (89) and (94)indicate that the FuelRates only depend on the attributes of buildingthermal conductivity UA^(Total) and HVAC system efficiencies with oneefficiency for the heating season and one efficiency for the coolingseason. These parameters are independent of weather conditions, occupanttemperature preferences, and internal gains. Furthermore,FuelRate^(Heating) and FuelRate^(Cooling) incorporate HVAC systemperformance, in contrast to thermal conductivity UA^(Total) which onlycharacterizes building shell performance. These properties satisfy theselection criteria outlined at the beginning of this section.

Determining FuelRate

The analysis uses energy load data to determine the FuelRate metric.FIG. 19 is a flow diagram 240 showing a method for determining fuelrates and Balance Point Temperatures with the aid of a digital computerin accordance with a further embodiment. The method 240 can beimplemented in software and execution of the software can be performedon a computer system, such as further described infra, as a series ofprocess or method modules or steps, in combination with intelligent“smart” metering, heating and cooling components, intelligent “smart”thermostats, and other devices that manage, control, monitor, andoperate energy consuming items.

This section demonstrates the methodology followed when natural gas isused for heating. Similar calculations are performed mutatis mutandisfor cooling, heating, or both when using other fuel or energy sources,as applicable. FuelRate is calculated using Equation (91). No on-siteaudit is performed. Equation (91) requires: (1) measured heating fuelconsumption over the period, (2) measured outdoor temperature data overthe same period, and (3) the Balance Point Temperature. Measured heatingfuel consumption and the Balance Point Temperature are typicallyunknown.

In addition, on-site power generation, which will typically bephotovoltaic (solar) power generation, although other forms of on-sitepower generation may apply, must be considered when reconstructing acustomer's energy load. Determining a customer's energy load withrespect to on-site power generation devolves into three cases. First,where a power utility customer only uses electricity supplied by thepower utility and does not produce any energy through on-site powergeneration means, the energy load will be based simply upon net(electric) load as measured over a set time period, such as metered by apower utility's on-site power meter for a monthly electricity bill.Second, where a customer also has an on-site solar power generationsystem (or other type of on-site power generation system) installed thatis separately metered by the power utility, the electric load will bebased upon the power utility-metered net electric load plus the powerutility-metered solar power generated during that same time period.Third, where on-site solar power generation is in place but is notmetered by the power utility (or is metered by the power utility but notadded into the net load), the electricity generated through solar powerwill need to be added to the power utility-metered net load to yield theenergy load in toto. Otherwise, the energy load will be underreporteddue to the exclusion of on-site solar power generation. Note that therecould also be other metering devices installed at a customer's building,such as a meter for natural gas, liquid propane, or water. Whereapplicable, the energy consumption based on a metered fuel might alsoneed to be factored into a customer's energy load in a manner similar tothe third case of unmetered on-site solar power generation.

On-site solar power generation can be determined by obtaining historicaldata chronicling on-site power generation over the same time period asnet load for electricity consumption (from, for instance, the powerutility), provided that the on-site power generation was separatelymetered though a meter within the customer's electricity circuits beforetheir (single utility-provided) electricity power meter or otherwisemeasured or tracked. Alternatively, the on-site power generation can beestimated by relying on a retrospective “forecast” of photovoltaic powergeneration created using, for instance, the probabilistic forecast ofphotovoltaic fleet power generation, such as described infra beginningwith reference to FIG. 1, by assuming a fleet size that consists of onlyone photovoltaic system using Equation (12). Other methodologies fordetermining on-site solar (or other) power generation are possible.

Energy that has been used specifically for interior environmentalconditioning (space heating or cooling), whether electricity orconsumable fuel, is not typically measured separately because powerutility customers generally have only a single power meter installed ontheir building for each energy or fuel source. A single power meter canonly measure net load for that type of energy and is unable todisaggregate load by individual component use. As a result, whenmeasured using a single power meter, the meter's measurement of net loadwill combine all downstream consumption for the devices within abuilding that rely on the energy provided through that power meter,regardless of component load type, including seasonal heating andcooling fuel consumption, as well as end-uses, such as water heating,cooking, and clothes drying. Adjustment to factor in on-site powergeneration may be required, as discussed infra.

Net load, as periodically measured through a power meter or similarsingle-point source, over a period of interest p is obtained (step 241).The net load data can be obtained from utility-metered energyconsumption data as generally maintained by a power utility for billingpurposes, or from other sources of data, such as a customer's utilitybill records. The net load could also be measured on-site, provided thatpower metering infrastructure is available and accessible, such asthrough the use of a smart meter.

If no on-site power generation is in use, the net load can be used asthe energy load for the building. However, net load will need to beadjusted (step 242) in situations where on-site power generation, suchas a photovoltaic system, is installed or another fuel is consumed, suchas natural gas, for space heating. Where no separate power meter hasbeen installed in the building for measuring on-site power generation,the “raw” net load obtained from the power utility or other source willeffectively disguise any power produced by an on-site power generationsource that has occurred downstream from the single power meter for onlymeasuring utility-supplied electricity. By providing electricity to thebuilding after the point at which the power utility measuresconsumption, on-site power generation decreases the net load observed bythe power utility for the building and effectively hides the overallenergy load. Actual energy load can only be correctly assessed bycombining the power meter-measured net load with any on-site generationsource-contributed energy to yield the energy load (step 242). On-sitesolar power generation can be obtained from either historical records,provided that on-site power generation was measured and recorded, or theon-site power generation can be estimated (retrospectively “forecast”)using the probabilistic forecast of photovoltaic fleet power generation,such as described infra beginning with reference to FIG. 1, by assuminga fleet size that consists of only one system using Equation (12). Othermethodologies for determining on-site power generation are possible

The measured outdoor temperature data for the customer's location overthe same period T _(p) ^(Outdoor) is also obtained (step 243), which canbe obtained from the SolarAnywhere® database service, cited supra, orother sources of meteorological data. The meteorological data could alsobe measured on-site, provided a temperature monitoring infrastructure isavailable.

The issue of the unavailability of space heating and space cooling fuelusages can be addressed by adding the average rate of other fuelconsumption in the heating period to Equation (92) to get average fuelfor all loads over the selected period to which the heating formulaapplies:

$\begin{matrix}{{AvgFuel}_{p} = {{\overset{\overset{Constant}{︷}}{- \left( {FuelRate}^{Heating} \right)}\overset{\overset{Varies}{︷}}{\left( {\overset{\_}{T}}_{p}^{Outdoor} \right)}} + \overset{\overset{Contant}{︷}}{{AvgOtherFuel} + {\left( {FuelRate}^{Heating} \right)\left( T^{{Balance}\mspace{14mu}{Point}} \right)}}}} & (95)\end{matrix}$Likewise, a similar approach is applied to a cooling period:

$\begin{matrix}{{AvgFuel_{p}} = {{{- \overset{\overset{Constant}{︷}}{\left( {FuelRate}^{Cooling} \right)}}\left( \overset{\overset{Varies}{︷}}{{\overset{¯}{T}}_{p}^{Outdoor}} \right)} + \overset{\overset{{Cont}{ant}}{︷}}{{AvgOtherFuel} - {\left( {FuelRate}^{Cooling} \right)\left( T^{BalancePoint} \right)}}}} & (96)\end{matrix}$Equations (95) and (96) can be expressed as a pair of lines in aslope-intercept form and can be expressed in a graph or other numericalrepresentation (step 244) to respectively represent the average fuelconsumptions for heating and cooling. Average outdoor temperature duringthe period is the independent variable (along the x-axis) and averagemetered fuel (for all loads) during the period is the dependent variable(along the y-axis). The FuelRate^(Heating) and FuelRate^(Cooling) arethen determined by finding the slopes of the lines respectivelycorresponding to Equations (95) and (96) (step 245). The slopes of thelines can be calculated using linear regression, or some othermethodology. Finally, the two Balance Point Temperatures are found (step246) by determining the outdoor temperature above which energyconsumption deviates from the lines representing heating and cooling.

As illustrated in FIG. 18, plotting the (x, y) results for each monthover several years generates points along the two lines. The pointsfollow a season-dependent linear trend, as predicted by Equations (95)and (96). Natural gas consumption depends on temperature only in theheating season. The home in this example does not use natural gas forspace cooling. The resulting slope of line 230 represents the FuelRatefor heating, as predicted by Equation (95).

Consistency Under Diverse Conditions

FuelRate is an objective measure of building performance. FuelRate canbe calculated using two readily-available, empirical data sources: (1)outdoor temperature; and (2) natural gas usage. The same methodology canbe applied to electricity consumption for cooling, electricityconsumption for electric space heating, fuel oil for heating, or anyother combination of fuels for heating or cooling, provided that thefuel consumption data and local outdoor temperature data are available.

The analysis has been applied to geographically separated locations todemonstrate consistency of results under diverse conditions. FIG. 20includes graphs depicting, by way of examples, average fuel consumptionversus average outdoor temperature for four configurations. In each ofthe graphs, the x-axis represents average outdoor temperature in degreesFahrenheit and the y-axis represents average natural gas consumption inBtu per hour (Btu/hr). Referring first to graph (a) in FIG. 20, thegraph depicts three years of monthly natural gas data for a home inAlbany, N.Y. This home had a natural gas boiler for heating. Referringnext to graph (b) in FIG. 20, the graph depicts two years of monthlynatural gas data for a home in Kirkland, Wash. This home had a naturalgas furnace. Referring next to graph (c) in FIG. 20, the graph depictsfive years of monthly natural gas data for home in Napa, Calif. Thishome had a natural gas furnace. Finally, referring to graph (d) in FIG.20, the graph depicts 90 days of daily electricity consumption for thesame home in Napa, Calif. after conversion to an electric heat pump andimprovements to the building shell. Importantly, the slopes of the linesin each graph are well-behaved across geographical regions and heatingfuels, which suggests that FuelRate can be obtained at any utility wheretotal fuel consumption is metered. Thus, the analysis methodology isbroadly applicable by region and fuel since all power utilities meterfuel consumption for billing purposes.

Effective R-Value

As discussed supra, the analysis produces two key empirical metrics,FuelRate^(Heating) and FuelRate^(Cooling), to characterize buildingperformance. This part applies results from the analysis and uses theFuelRate metric to derive a building metric that is an intuitive,standard metric that can be used for all buildings, called the EffectiveR-Value (R^(Effective)).

Derivation

Fourier's Law characterizes thermal conduction through a material.Fourier's Law states that the flow of heat per unit area is proportionalto the temperature gradient across the material. Building analysiscommonly expresses this property in terms of thermal resistance, or“R-value.”

Fourier's Law can be written for a surface that has an area of A^(Total)as:

$\begin{matrix}{\overset{.}{Q} = \frac{A^{Total}\Delta\; T}{R}} & (97)\end{matrix}$

An equivalent Effective R-Value for a building that represents theoverall resistance to heat flow can be expressed as an overall heattransfer coefficient that can be used to calculate total heat transferthrough all surfaces, for instance, windows, walls, and so forth, thatmake up the total building surface area A^(Total):

$\begin{matrix}{{\overset{.}{Q} = {(U)\left( A^{Total} \right)\Delta\; T}}{{where}\text{:}}} & (98) \\{R^{Effective} = \frac{1}{U}} & (99)\end{matrix}$

Thus, Effective R-Value can serve as an alternative metric forcharacterizing overall building performance, particularly as most peopleare already familiar with the concept of an R-value. Effective R-Valueis also positively correlated with efficiency, where a higher R-valuecorrelates to a better thermally efficient building.

Solving Equation (89) for (1/U) and substituting for Effective R-ValueR^(Effective) yields:

$\begin{matrix}{R^{Effective} = \frac{A^{Total}}{{FuelRate}^{Heating}*\eta^{Heating}}} & (100)\end{matrix}$Equation (100) indicates that Effective R-Value is calculable from threedirectly measurable inputs: (1) building surface area (A^(Total)), (2)FuelRate^(Heating), and (3) HVAC heating efficiency (η^(Heating)).

Note that Equation (100) can be stated in a more generalized form forwhen there are two sources of heating. For example, with a natural gasfurnace, heat is delivered directly by burning natural gas and also bythe waste heat from the fans used to move heat around the home. Thus,the heating efficiencies of the two different systems that deliver heatto the building must be separately applied, that is, determine thenatural gas FuelRate and apply the HVAC efficiency of the natural gassystem, then determine the electricity FuelRate and apply the efficiencyof the waste heat from electricity, which would be 100%.

Estimating a Building's Surface Area

Calculating an Effective R-Value for a house requires the thermalconductivity UA^(Total), the building's surface area, and heating orcooling system efficiency, η^(Heating) and η^(Cooling), respectively. Abuilding's surface area can be estimated based on floor area and heightsand number of floors, which are values that are typically availablethrough tax assessor records. This estimation assumes that the buildinghas a square base.

Let A represent the floor area of a building, F represent the number offloors, and H represent the height per floor. The area per floor equalsthe total floor area divided by the number of floors.

Surface Area of Ceiling and Floor

The surface area of the ceiling equals the area per floor. The surfacearea of the floor also equals the area per floor. Thus, the combinedsurface area of the ceiling and floor equals

$2 \times {\frac{A}{F}.}$

Surface Area of Walls

The length of one side of a floor equals the square root of the area perfloor, or

$\sqrt{\frac{A}{F}}.$The surface area of the four walls equals four times height times thenumber of floors times the length of a side, that is,

$\left. {4{HF}\sqrt{\frac{A}{F}}} \right).$

Total Surface Area of Ceiling, Floor, and Walls

The total surface area A^(Total) equals the sum of the surface areas ofthe ceiling, floor and walls:

$\begin{matrix}{A^{Total} = {{{2\left( \frac{A}{F} \right)} + {4{HF}\sqrt{\frac{A}{F}}}} = {{2\left( \frac{A}{F} \right)} + {4H\sqrt{AF}}}}} & (101)\end{matrix}$

Ratio of Surface Area to Floor Area

To find the ratio of the surface area to the floor area, divide Equation(101) by A:

$\begin{matrix}{{{Ratio}\mspace{14mu}{of}\mspace{14mu}{Surface}\mspace{14mu}{Area}\mspace{14mu}{to}\mspace{14mu}{Floor}\mspace{14mu}{Area}} = {\frac{2}{F} + {4H\sqrt{\frac{F}{A}}}}} & (102)\end{matrix}$

Translation of Results

This section derives Effective R-Values for representative homes basedon results from JOHN RANDOLPH & GILBERT M. MASTERS, ENERGY FORSUSTAINABILITY: TECHNOLOGY, PLANNING, POLICY (Island Press 2008) anddemonstrates that the Effective R-Value is typically R-5 for olderhouses, R-8 for new code-built houses, and R-17 for super insulatedhouses (R-values in units of hr-° F.-ft²/Btu). Randolph and Masterscreated an index called the Thermal Index (Id. at 244), derived bymultiplying a building's thermal conductivity UA_(Total) by 24 hours anddividing by the building's floor area:

$\begin{matrix}{{{Thermal}\mspace{14mu}{Index}} = \frac{{UA}^{Total}*24}{{Floor}\mspace{14mu}{Area}}} & (103)\end{matrix}$

The relationship between Effective R-Value and Thermal Index can bedetermined by combining Equations (99) and (103). Substituting

${UA}^{Total} = \frac{A^{Total}}{R^{Effective}}$into Equation (103) and solving for Effective R-Value equals:

$\begin{matrix}{R^{Effective} = {\left( \frac{24}{{Thermal}\mspace{14mu}{Index}} \right)\left( \frac{A^{Total}}{{Floor}\mspace{14mu}{Area}} \right)}} & (104)\end{matrix}$

Randolph and Masters present predicted numbers for their Thermal Index(Id. at 245). Assuming that these values are for single-story, 1,500 ft²houses, the conversion to Effective R-Value is presented Table 3.

TABLE 3 Thermal Effective Index R-Value Older houses 15 5 New code-builthouses 8 8 Super insulated houses 4 17

Example Calculation of Effective R-Value

Consider an example Effective R-Value calculation. In graph (c) in FIG.20, the graph depicts natural gas consumption data for the home in Napa,Calif. Natural gas furnaces require electricity to operate the airdistribution fans. As a result, some electricity consumption during theheating season is temperature-related. FIG. 21 is a graph depicting, byway of example, average electricity consumption versus average outdoortemperature for the sample home in Napa, Calif. based on monthly data.The x-axis represents average outdoor temperature in degrees Fahrenheit.The y-axis represents average electricity load in kW.

As discussed supra, the slope of the lines 240 and 250 represents theFuelRate metric. Graph (c) in FIG. 20 implies that the home wouldconsume 46,300 Btu per hour of natural gas when the outdoor temperaturewas 10° F. and 4,500 Btu per hour when the outdoor temperature was 55°F., where line 240 has a slope of 929 Btu/hr-° F. Thus, the natural gasFuelRate is 929 Btu/hr-° F. FIG. 21 implies that the electricityFuelRate is 24 Watts/° F., which converts to 82 Btu/hr-° F. (in Imperialunits).

Assume that the natural gas furnace in the sample house had a 70%efficiency, including duct losses, and that the waste heat from theelectric fans had a 100% efficiency, that is, all of the heat generatedby operation of the fans heated the home. Table 4 illustrates that thenatural gas furnace delivered 650 Btu/hr-° F. and, when combined withthe waste heat from electricity, the total heat delivered was 732Btu/hr-° F.

The usable floor space (obtainable from public tax records) can be usedto calculate A^(Total) using the methodology for estimating surface areafrom building square footage and floor area, described supra. Thus, thistwo-story, 2,871 ft² house has an estimated surface area of 5,296 ft².The house has an Effective R-Value of R-7.2.

TABLE 4 HVAC Delivered Fuel Rate Efficiency Heat Natural Gas 929Btu/hr-° F.  70% 650 Btu/hr-° F. Electricity  82 Btu/hr-° F. 100%  82Btu/hr-° F. 732 Btu/hr-° F. Surface to Floor Area Floor Ratio SurfaceArea Building Area 2,871 sq. ft 1.84 5,296 sq. ft. Effective R-Value   7.2

Effective R-Value as a Building Standard

The Effective R-Value is a promising metric for characterizing buildingperformance. Policy makers could use this value as the basis of abuilding standard by requiring that buildings meet a specified numericR-Value. There are several advantages of using Effective R-Value as astandard, as follows.

Effective R-Value Normalizes Building Performance

First, Effective R-Value normalizes building performance for weatherconditions, building configuration, and consumer preferences. EffectiveR-Value is independent of weather conditions since the underlyingFuelRate calculation removes the effect of weather conditions. Thisaspect of Effective R-Value allows buildings in different regions to bedirectly compared.

For example, the Effective R-Value can be compared to the Passive Housestandard (stated in Btu/hr-ft²), as discussed in more detail infra, bydividing by floor area and converting to Watts per square meter. Thegeneral form of the equation is that the house meets the Passive HouseStandard if the following condition is satisfied:

$\begin{matrix}{{{Passive}\mspace{14mu}{Standard}} \geq \left\lbrack \frac{\left( \frac{{Surface}\mspace{14mu}{Area}}{R\text{-}{Value}} \right)\left( {T^{Balance} - T^{Design}} \right)}{{Floor}\mspace{14mu}{Area}} \right\rbrack} & (105)\end{matrix}$Rearranging Equation (105) to solve for the R-Value yields:

$\begin{matrix}{{R\text{-}{Value}} \geq {\left( \frac{1}{{Passive}\mspace{14mu}{Standard}} \right)\left( \frac{{Surface}\mspace{14mu}{Area}}{{Floor}\mspace{14mu}{Area}} \right)\left( {T^{Balance} - T^{Design}} \right)}} & (106)\end{matrix}$

The Passive House standard is 10 W/m². Using the correct conversionfactor, which is

${{\left( \frac{3.412\mspace{14mu}{Btu}\text{/}h}{W} \right)\left( \frac{1\mspace{14mu} m}{3.281\mspace{14mu}{ft}} \right)^{2}} = {{0.3}17\frac{{{Btu}\text{/}h} - m^{2}}{W - {ft}^{2}}}},$the Passive House standard is equivalent to

$3.17\frac{{Btu}\text{/}h}{{ft}^{2}}$The ratio of Surface Area to Floor Area equals

$\frac{2}{F} + {4H{\sqrt{\frac{F}{A}}.}}$Making the correct substitutions, an R-Value

$\left( {{in}\mspace{14mu}{units}\mspace{14mu}{of}\mspace{14mu}\frac{{ft}^{2} - {{^\circ}\mspace{14mu}{F.}}}{{Btu}\text{/}h}} \right)$satisfies the Passive House standard if:

$\begin{matrix}{{R\text{-}{Value}} \geq {\left( {{0.3}153} \right)\left( {\frac{2}{F} + {4H\sqrt{\frac{F}{A}}}} \right)\left( {T^{Balance} - T^{Design}} \right)}} & (107)\end{matrix}$

The Passive House Standard for a two story, 2,871 ft² house with floorheights of eight feet, a Balance Point Temperature of 58° F., and DesignTemperature of 33° F. is satisfied if the R-Value of the house exceeds14.5, that is,

$\left. {{(0.3153)\left( {\frac{2}{2} + {4*8\sqrt{\frac{2}{2.871}}}} \right)\left( {{58^{\circ}} - {33{^\circ}}} \right)} = {14.5\frac{{ft}^{2} - {{^\circ}\mspace{14mu}{F.}}}{{Btu}\text{/}h}}} \right\rbrack.$

In addition, Effective R-Value normalizes results based on surface area,not floor area, which removes the effect of building configuration.Thus, the Effective R-Value for a single-story home can be directlycompared to the Effective R-Value for a multi-floor apartment building.

Finally, Effective R-Value removes the effect of consumer preferences,particularly average indoor temperature. Consumer behavior affects totalfuel consumption. The effect however, occurs through the Balance PointTemperature and not the FuelRate. Note that the Effective R-Valueaffects Balance Point Temperature, but not vice-a-versa.

Effective R-Value is Based on Measured Data

Effective R-Value is based on measured data, as demonstrated herein.This aspect of Effective R-Value means that compliance of a building canbe verified based on measured energy consumption.

Effective R-Value Supports Detailed Energy Audits

Effective R-Value is an empirically-derived number that supportsdetailed energy audits. The sum of the losses reported by a detailedenergy audit, normalized by surface area, must equal the inverse ofEffective R-Value. If the results do not match, something is incorrectin either the detailed energy audit, total surface area, or assumed HVACefficiency. This leads to the approach where the specific surface areaR-Values can be confidently estimated without requiring an on-sitevisit.

Effective R-Value Accurately Predicts Fuel Consumption when Combinedwith Other Parameters

Effective R-Value accurately predicts fuel consumption when combinedwith surface area, HVAC Efficiency, and the Balance Point Temperature.Fuel consumption can therefore be calculated with four building specificnumbers, one of which depends on consumer behavior, and weather data.This aspect of Effective R-Value allows future fuel consumption in lightof investments that improve a home's Effective R-Value to be accuratelypredicted and also allows long-term fuel consumption to be predicted bycombining results with previous weather data sets.

Effective R-Value May be Meaningful to Average Consumers

As a standardized metric, Effective R-Value may be meaningful to averageconsumers. Building energy efficiency experts often use terms that havelittle meaning to typical consumers. However, stating results asEffective R-Values will provide consumers with a metric of comparison.For example, if a consumer has a home with an Effective R-Value of R-6,that value can be compared to an R-13 value associated with 4″ offiberglass insulation in a wall with 2×4 studs.

Comparison to Existing Standards

Effective R-Value is a useful metric for several reasons. First, themetric normalizes building performance for weather conditions andbuilding configuration, including floor area and number of floors.Second, the metric is independent of consumer temperature comfort level.Third, the metric is based on measured, rather than modeled, data.Fourth, the metric supports detailed energy audits. Fifth, the metriccan be used to predict annual fuel consumption when combined with energyinvestment scenarios, including on-site photovoltaic power generation.Finally, the metric may be an intuitive metric for average consumers.

Effective R-Value can be used to support or improve existing buildingstandards. Consider a few of the standards.

Zero Net Energy (ZNE)

A building meets the ZNE standard if renewable energy sources, such asphotovoltaic power generation, produce enough energy to supply annualconsumption. To determine whether a house meets the ZNE standard, simplysums net energy consumption, including electricity and natural gas, overthe course of a year and the house is ZNE if the total net energy equalszero or less.

The ZNE standard provides no information about the quality of thebuilding shell, the efficiency of the HVAC system, or the energyrequired to power the HVAC system in the winter and summer. Homeownerscan simply oversize photovoltaic generation to account for inefficientbuilding shells or HVAC systems. Furthermore, ZNE homes are not withoutemissions, such as occurs with natural gas consumption.

The Effective R-Value could be used to characterize existing buildingsprior to the installation of photovoltaic power generation to helpbuilding owners evaluate the trade-off between building shell and HVACinvestments versus incremental photovoltaic power generation capacity.Both investment options could result in a ZNE home, but the analysisusing Effective R-Value could lead to the most cost-effective pathtowards meeting the ZNE standard.

Home Energy Rating System (HERS)

HERS is a common building performance rating system; a HERS index of 100corresponds to the energy use of the “American Standard Building,” whilea HERS index of 0 indicates that the building uses no net purchasedenergy. The HERS rating system informs homeowners on how their homescompare to other homes, yet the system does not explicitly guideconsumers how to improve their home's energy performance. Further,results are primarily based on modeled results, rather than empiricallymeasured values. By incorporating the Effective R-Value into theanalysis, building owners could make better informed decisions about howto improve the HERS ratings of their buildings.

California Title 24 Energy Code

The California Title 24 energy code takes a prescriptive approach tobuilding standards. A prescriptive approach may be useful when guidingconsumers about which investments to make; however, this approach doesnot provide a good measure of the overall thermal performance of anexisting building. Policy makers could use the Effective R-Value as analternative building standard that allows for such variations.Furthermore, a building standard based on Effective R-Value would lenditself to measurement and verification using actual usage data.

Passive House

The Passive House standard was developed in Germany in the 1990s and hasbecome one of the most aggressive thermal home performance standards inthe world. The Passive House standard requires that a house meet fourrequirements covering space heating (and cooling, if applicable) energydemand(s), renewable primary energy demand, airtightness, and thermalcomfort. The Passive House standard is appealing in that the standardprovides a Yes or No answer as to whether a home is compliant. However,the standard does not provide a fundamental answer on building shelleffectiveness. Moreover, the Passive House standard islocation-dependent and measured as an annual energy (demand) consumptionor a peak demand value.

The analysis discussed herein uses building surface area, rather thanfloor area, because the underlying equations governing heat transfer arebased on the area through which heat passes, considerations that are notobserved under the Passive House standard, which defines maximum usageper unit of floor area.

Complete Virtual Energy Audit

Here, building performance is remotely analyzed using customer energyload, collected and aggregated in either hourly or monthly intervals,and externally-supplied meteorological data. These data are used todisaggregate building thermal load into individual categories ofcomponent loads, such as heating, cooling, and constant “always-on”baseloads, which can provide valuable insights into consumer behaviors.For instance, building thermal loads are temperature-driven and tend todefine a utility's system peak, which is a crucial metric for balancingpower generation and consumption.

The analysis discussed herein, also referred to as the Virtual EnergyAudit, encompasses both thermal and non-thermal loads, that is, loadsother than heating and cooling. This section provides a methodology fordisaggregating electrical loads into four component loads, heating,cooling, baseload, and other. The methodology can take advantage ofadvanced metering infrastructure (AMI, or simply, “smart metering”), asavailable, including the detection of baseloads and days when a buildingis unoccupied.

The Virtual Energy Audit produces the following information:

-   -   Building-specific, objective parameters useful in modeling the        building's energy consumption.    -   “Effective R-Value,” an intuitive comparative metric of overall        building thermal performance.    -   Disaggregation of electrical loads into heating loads, cooling        loads, baseloads, and other loads.        This information can be used by the planners and operators of        power utilities to help assess on-going and forecasted power        consumption and in adjusting or modifying the generation or        procurement of electric power, as well as for other purposes of        concern to a power utility or other related parties.

Bundling Technologies with Solar: The Solar+ Home

Solar photovoltaic system installations have grown exponentially overthe last decade with these systems being bundled with othertechnologies. For example, solar power generation is being increasinglybundled with power storage (Solar+Storage). Solar power generation isalso being bundled with electric vehicles (Solar+EV) and with efficiency(Solar+Efficiency). A combination of all three technologies could betermed “Solar+Storage+EV+Efficiency.” However, the combination does notguarantee a beneficial load profile from the perspective of the powerutility because the way that these technologies impact a power griddepends upon the dispatch method. As used herein, homes that employ andoperate these technology bundles for the mutual benefit of the customerand the power utility will be called “Solar+homes.”

The Solar+Home Approach

A Solar+home is created by combining multiple technology components.FIG. 22 is a block diagram showing, by way of examples, Solar+hometechnology blends. In this example, there are three broad categories ofinvestments. In the first category (shown on the left), a Solar+home hasa primary supply of electrical energy that is solar photovoltaic powergeneration, either rooftop or community photovoltaic, that is,photovoltaic power generation situated on the Solar+home or from apooled shared solar photovoltaic system. In the second category (shownin the middle), a Solar+home incorporates demand-side investments toreduce and electrify consumption. In the third category (shown on theright), a Solar+home matches the investments in supply (solarphotovoltaic power generation) with the investments in demand, which canbe accomplished by using demand response/load control, load shifting,customer-sited or utility-sited (energy) storage, or a combination ofthe foregoing options.

Prototype Solar+Home Monitoring

For empirical purposes, a Solar+home approach was applied to an existingresidential retrofit home in Napa, Calif. This sample home has 2,871 ft²of floor area. Prior to any investments, a manual, on-site energy auditdetermined that the home had a HERS rating of 125, which meant that thehome consumed more energy than the “American Standard Building.”

Measured Disaggregated Energy Consumption

Monitoring of detailed end-use energy consumption in the sample home hasbeen performed since June 2014. FIG. 23 is a bar chart showing, by wayof example, three-and-a-half years of consumption and production data bycircuit for the sample home. Table 5 presents the measured consumptionand production data values. In FIG. 23, the x-axis represents theconsumption and production data grouped into applicable periods of timeand the y-axis represents energy in kWh. Each reported year starts onJuly 1. The final year only contains six months of data with resultsscaled by multiplying by two. The data acquisition system monitored tencircuits, (1) refrigerator, (2) heating (using electric heating in the2014-2015 and 2015-2016 heating seasons, then switching to a heat pumpin the 2016-2017 heating season) and cooling, (3) cooking (threeseparate circuits for cooktop, oven, and microwave), (4) laundry (washerand dryer combined), (5) dishwasher, (6) heat pump water heater, (7) EV,and (8) solar photovoltaic. In addition, detailed temperature, CO₂, andhumidity measurements were recorded at five locations throughout thehome (upstairs, downstairs, attic, garage, and outdoors) since April2015.

Note that there is a wide variation in electricity used for heatingacross each of the years; the first and third years had approximatelythe same amount of consumption, while consumption in the second year wasmore than double that of the first year. Also, there was a notableincrease in the amount of energy used for lights and miscellaneous itemsin 2016-17. The variation in electricity used for heating isparticularly surprising because electric resistance was the heat sourcefor the first two years, while a high efficiency mini-split heat pumpwas the heat source in the third year, which suggests that the change inconsumption was weather-related.

TABLE 5 Jul. 1, 2014 - Jun. 30, 2015 Jul. 1, 2015 - Jun. 30, 2016 Jul.1, 2016 - Dec. 31, 2016 Jul. 1, 2017 - Dec. 31, 2017 Cons. Prod. Cons.Prod. Cons. Prod. Cons. Prod. Refrigerator 512 467 470 523 Lights &Misc. 1,460 1,499 2,080 1,964 Space Heater 1,780 3,838 1,228 909 Cooling0 50 27 49 Cooking 527 385 457 494 Laundry* 146 111 141 136 Dishwashing224 184 209 236 Water Heating 727 579 594 535 EV 2,340 1,799 3,935 2,119Solar 8,108 7,814 7,725 7,786 *Natural gas consumed for laundry isexcluded. It equals about 24 therms per year.

Virtual Energy Audit Procedure

The Virtual Energy Audit can be performed over a shorter or longerperiod or periodically to determine changes in building performance.FIG. 24 is a flow diagram showing a method 260 for performing powerutility remote consumer energy auditing with the aid of a digitalcomputer in accordance with a further embodiment. The method 260 can beimplemented in software and execution of the software can be performedon a computer system, such as further described infra, as a series ofprocess or method modules or steps, in combination with intelligent“smart” metering, heating and cooling components, intelligent “smart”thermostats, and other devices that manage, control, monitor, andoperate energy consuming items. In addition, by way of examples, FIG. 25includes graphs showing the results for each step of the method 260 ofFIG. 24 for the sample house.

As an initial step, an energy load for the building can be assessed(step 261). Data for assessing the energy load can be obtained fromutility-metered energy consumption data, such as generally maintained bya power utility for billing purposes, or from other sources of data,such as a customer's utility bill records. The energy load data couldalso be measured on-site, provided that power metering infrastructure isavailable, such as through a smart meter. Note that at this stage, theenergy load is preliminary and subject to adjustment in situations whereon-site power generation is use, as discussed infra.

Second, the stability of the energy load is verified (step 262). Energyload stability can be assessed by evaluating cumulative energy loadversus time, then scanning the results for discontinuities orirregularities. For instance, the assessment can be made using a simpletwo-dimensional graph. Referring to graph A in FIG. 25, the x-axisrepresents dates occurring during the period of interest, Sep. 16, 2016to Sep. 15, 2017 and the y-axis represents cumulative fuel consumptionin kWh. The energy load over the period of interest reflects the notableincrease in the amount of energy used for lights and miscellaneous itemspreviously observed in 2016-17.

Referring back to FIG. 24, this step identifies significant changes inthe energy load that could invalidate analytical results; adjustmentscan be made to the energy load as necessary to address any significantchanges identified. For example, a customer could have added a new roomto the home, installed a photovoltaic power generation system, purchasedan electric vehicle, or switched from natural gas to heat pump spaceheating. Investments that occur in the middle of an analytical timeframe may require adjustments, such as restricting the analysis to atime frame before or after the investment.

Third, the energy load, as verified and adjusted to address anysignificant changes, may need to be further adjusted to factor in anyon-site power generation, which will typically be photovoltaic (solar)power generation. Where no on-site power generation is installed, thecustomer's energy load is simply based upon net energy (electric) loadas measured over a set time period, such as metered by a power utility'son-site power meter for a monthly electricity bill and no adjustment tothe energy load is required for present analytical purposes.

Similarly, if on-site solar (or other) power generation is in use (step263), and a separate power meter for the on-site solar power generationsystem is installed (step 264), the energy load will not need to beadjusted for present analytical purposes, provided that the customer'spower utility is monitoring both the building's electric power meter andthe solar power meter and that the data for the energy load is providedby the utility as the sum of utility-supplied electricity and on-sitepower generation. If the power utility provides individual values forthe separately metered utility-supplied electricity and on-site powergeneration, the energy load required for present analytical purposes canbe readily determined by just adding these two values.

An adjustment to the energy load may be needed if there is no separatesolar power meter in use (step 264). The adjustment is made by addingthe on-site power that was generated over the same period to the energyload (step 265) to yield the energy load required for present analyticalpurposes. Typically, on-site power will be generated through solar,although other forms of on-site power generation may apply. On-sitesolar power generation can be obtained from either historical records,provided that on-site power generation was measured and recorded, or theon-site power generation can be estimated by retrospectively“forecasting” power generation, for instance, by using a probabilisticforecast of photovoltaic fleet power generation, such as described infrabeginning with reference to FIG. 1, by assuming a fleet size thatconsists of only one system in Equation (12). Other methodologies fordetermining on-site power generation are possible. Note that this sameadjustment would need to be made to the energy load for any other typesor additional sources of on-site power generation, including fuel cells,natural gas generators, diesel generators, and so forth.

Fourth, the baseload is determined (step 266). The baseload can beestimated using the total customer electric load formed into a timeseries, such as described in commonly-assigned U.S. Pat. No. 10,747,914,issued Aug. 18, 2020, the disclosure of which is incorporated byreference. The time series is then expressed as a frequency distributionas a function of average periodic demand for power, where the averageperiodic demand for power that most frequently occurs during the periodof observation is identified as the baseload for the building. Referringto graph B in FIG. 25, the x-axis represents total metered load in kWand the y-axis represents percentage of load distribution with theaverage periodic demand occurring at 0.16 kW.

Referring back to FIG. 24, this step identifies the mode, that is, themost common reading, of electrical usage as the “always on” reading,which is different than using the minimum reading, which would subjectthe result to error under certain conditions, such as when a poweroutage occurs during the time frame. As discussed herein, annualbaseload consumption was determined by multiplying the baseload rate(kWh per hour) by 8760, the number of hours in a year, although otherways to annualize baseload consumption are possible.

Fifth, days when either the house was unoccupied or only partiallyoccupied are eliminated from the analysis time frame (step 267). Theanalysis discards these days because they can introduce error into thethermal analysis. Referring next to graph C in FIG. 25, the x-axisrepresents dates occurring during the period of interest, Sep. 16, 2016to Sep. 15, 2017 and the y-axis represents total metered load in kW.There were six periods identified that contain days when either thehouse was unoccupied or only partially occupied.

Referring back to FIG. 24, this step eliminates factors that reduceinternal gains. For example, occupants may turn down the thermostat whenthey leave the house for an extended period, plus there is less wasteheat due to reduced electric device usage and less occupant body heat.Unoccupied days can be identified by comparing daily load to dailybaseload, that is, the instantaneous baseload times 24 hours. A buildingcan be classified as being unoccupied if the daily load is less than 120percent of daily baseload, although other percentage comparisons arepossible. In addition, the day before and the day after an unoccupiedday can both be classified as partially occupied days and excluded toreflect the departure and return of the occupants as having occurredsometime during the course of those days.

Sixth, the FuelRates for heating and cooling and the Balance PointTemperatures are calculated (step 268), as described supra withreference to FIG. 19. Total non-HVAC loads are also calculated bycomparing average load over periods that contain multiple consecutivedays (with possible gaps due to unoccupied or partially occupiedconditions) to average temperature during the periods withexternally-supplied meteorological data to analyze each customer'sbuilding performance. Referring next to graph D in FIG. 25, the x-axisrepresents average outdoor temperatures in degrees Fahrenheit and they-axis represents average metered load in kW. The heating line has aslope (FuelRate) of 0.053 kW/° F. and a Balance Point Temperature of 60°F. The cooling line has a slope (FuelRate) of 0.028 kW/° F. and aBalance Point Temperature of 67° F. The dashed baseload line is 0.16 kWand the solid horizontal other load line is 0.39 kW. These values areindependent of temperature.

Referring back to FIG. 24, the two Balance Point Temperatures are foundby determining the outdoor temperature above which energy consumptiondeviates from the lines representing heating and cooling, as discussedsupra with reference to FIG. 18, in which the average amount of naturalgas consumed for heating (or average amount of electricity for cooling)are plot against the average outdoor temperature to form lines.

Seventh, an average daily outdoor temperature frequency distribution isgenerated (step 269) over a year. Other time frames could be used.Referring to graph E in FIG. 25, the x-axis represents average outdoortemperatures in degrees Fahrenheit and the y-axis represents temperaturedistribution in degrees Fahrenheit. The graph presents the average dailyoutdoor temperature frequency distribution for the sample home over theperiod of interest.

Referring back to FIG. 24, annual consumption for heating and coolingwas calculated by combining these frequencies of average daily outdoortemperature with the FuelRates for heating and cooling and the BalancePoint Temperatures.

Finally, the individual component loads are calculated by disaggregatingthe results from the previous steps (step 270). Equations (95) and (96),discussed supra, can be expressed as lines in a slope-intercept formwhose slopes respectively represent the average fuel consumptions forheating and cooling. In turn, based upon these values in combinationwith the load data and average daily outdoor temperatures, plus theBalance Point Temperatures as determined in this methodology, Equations(95) and (96) can be solved for the average fuel consumption for otherloads AvgOtherFuel value. By subtracting out the baseload alsodetermined in this methodology, the average fuel consumption for otherloads, that is, not for heating load, cooling load, or baseload, can befound. Referring to graph F in FIG. 25, the y-axis represents annualfuel consumption in kWh per year (kWh/yr), which presents a simplifiedset of results that the power utility could share with the customer tohelp promote understanding of the relative amounts of electricityconsumption.

On-site solar power generation can be found using Equation (72)(discussed in detail supra in the section entitled, “PhotovoltaicProduction”) if the total customer electric load and all individual loadcomponents at a given time interval are known. Equation (72) cantherefore be used to corroborate a retrospective “forecast” of on-sitesolar power generation in situations where historically measured solarpower generation data is not available.

Validation Using Measured Electricity Consumption and Temperature Data

While the Virtual Energy Audit is based on total energy consumption,detailed end-use consumption data on the Solar+home was collected andevaluated to empirically validate the results of the Virtual EnergyAudit. FIG. 26 includes graphs showing, by way of examples, a comparisonof results for the sample home obtained through the Virtual Energy Auditand measured energy consumption data. The x-axes group the results forthe Virtual Energy Audit and measured energy consumption data. They-axes represent annual fuel consumption in kWh per year (kWh/yr). GraphA in FIG. 26 presents outdoor temperature data that was measured on-siteand demonstrates that the Cooling and Heating categories comparecomparably. Baseloads were not measured separately. Graph B in FIG. 26presents the preceding analysis using outdoor temperature data obtainedfrom the SolarAnywhere® database service, cited supra, to demonstratethat the audit could be performed without on-site temperaturemeasurement. The results are nearly the same as when the analysis usedon-site outdoor temperature data.

Key Performance Parameters

The preceding analysis was repeated using three-and-a-half years of datafor the Solar+home. Table 6 presents the key parameters. There aretypically six parameters for each period or year. The 2008-2013 periodincludes results for both electricity and natural gas because the househad a natural gas furnace and natural gas water heating during thattime. The subsequent periods do not include natural gas because thespace heating and water heating were all electric. The house had no airconditioning in the 2014-2015 and 2015-2016 seasons.

TABLE 6 Thermal Performance Heating Cooling Fuel Fuel Non-HVAC LoadConsumption Balance Point Consumption Balance Point Year Total BaseloadRate Temperature Rate Temperature ′08 - ′13 2,750 Btu/h & 0.35 kW 929Btu/h-° F. & 58° F. 0.75 kW 0.024 kW/° F. ′14 - ′15 0.36 kW 0.12 kW0.129 kW/° F. 58° F. ′15 - ′16 0.38 kW 0.12 kW 0.137 kW/° F. 58° F.′16 - ′17 0.39 kW 0.16 kW 0.053 kW/° F. 60° F. 0.028 kW/° F. 67° F. ′17H2 0.39 kW 0.18 kW 0.039 kW/° F. 58° F. 0.019 kW/° F. 65° F.

Validation Using Equipment Ratings

The results were also validated by comparing the implied thermalconductivity derived using the heating and cooling fuel rates andequipment ratings. The Solar+home used resistance heating in the2014-2015 and 2015-2016 seasons. The home used a ductless mini-splitheat pump for heating and cooling in the 2016-2017 and 2017 H2 seasons.The ductless mini-split heat pump was a Ductless Aire Model DA1215-H1-0,manufactured by DuctlessAire, Columbia, S.C., with an input voltage of115 volts and heating and cooling capacity of 12,000 Btu/hour. Accordingto the manufacturer's website, the unit has a SEER rating of 15 and anHSPF of 8.2.

SEER is the ratio of BTU cooling output over a typical cooling season towatt-hours of electricity used. HSPF is the ratio of BTU heat outputover a typical heating season to watt-hours of electricity used. Both ofthese metrics have units of BTU/Watt-hr. Note that the actual heatingand cooling seasons can differ from the typical seasons used to developthe rating standards.

The Solar+home's thermal conductivity UA_(Total) equals FuelRate (kW per° F.) times the HVAC system efficiency, including ducting losses. Thishome has no ducting losses because the heat injection or removal takesplace directly through the wall without passing through ducts, so theHVAC system efficiency is the same as the equipment efficiency and iseither the SEER or the HSPF rating, depending on season. Table 7presents the estimated thermal conductivity using the manufacturer's 15SEER rating and 8.2 HSPF rating.

TABLE 7 Fuel HSPF or Thermal Rate SEER Conductivity Heating 53 W/° F. 8.2 Btu/Wh 435 Btu/h-° F. Cooling 28 W/° F. 15.0 Btu/Wh 420 Btu/h-° F.

The heating or cooling fuel consumption (in W/° F.) times HSPF (forheating) and SEER (for cooling) rating should result in the same thermalconductivity. Electricity consumption in the heating season with an HSPFof 8.2 implies a thermal conductivity of 435 Btu/h-° F. Electricityconsumption in the cooling season combined with a SEER rating of 15.0implies a thermal conductivity of 420 Btu/h-° F. The two results arewithin 4% of each other, which provides a separate confirmation of thevalidity of using the Virtual Energy Audit to determine a building'sthermal conductivity by combining either heating or cooling fuelconsumption rate with the corresponding HVAC system efficiency.

Effective R-Value

Table 4, supra, showed that the Solar+home had an Effective R-Value of7.2 prior to any investments. Table 8 repeats the analysis for theSolar+home after improvements to the building shell and the installationof the mini-split heat pump (9/16/16 to 9/15/17). The heatingrequirement of 0.053 kW/° F. is converted to 181 Btu/hr-° F. to findthat the Solar+home has an Effective R-Value of R-12.2.

TABLE 8 Fuel Delivered Consumption Efficiency Heat Electricity 181Btu/hr-° F. 240% 435 Btu/hr-° F. 435 Btu/hr-° F. Surface to Floor AreaFloor Ratio Surface Area Building Area 2,871 sq. ft. 1.84 5,296 sq. ft.Effective R-Value   12.2

The homeowner made additional investments to the Solar+home after the2016-2017 season, including upgrading the windows and installing awell-insulated new front door. FIG. 27 is a chart showing, by way ofexamples, the Effective R-Values of the sample house before upgrades,after most upgrades, and after all upgrades. Notice that the houseachieves approximately the value of a super-insulated home. This chartillustrates how a utility could use the Virtual Energy Audit to trackchanges in building performance over time and provide measurement andverification of building upgrades.

Virtual Energy Audit System

The Virtual Energy Audit methodology can be performed with theassistance of a computer, or through the use of hardware tailored to thepurpose. FIG. 28 is a system 330 for performing power utility remoteconsumer energy auditing with the aid of a digital computer 331 inaccordance with a further embodiment. A computer system 331, such as apersonal, notebook, or tablet computer, as well as a tablet computer,smartphone, or programmable mobile device, can be programmed to executesoftware programs 332 that operate autonomously or under user control,as provided through user interfacing means, such as a monitor, keyboard,and mouse. The computer system 331 can be on-site or could be remotelyoperated by a power utility or other endeavor.

The structure 338 is supplied electricity 342 from a power utility (notshown) over an electric power line. Consumption of the electricity 342is measured or monitored by the power utility through a conventionalinterval or time-of-use power meter (not shown) or a power meteringinfrastructure 336, which generally includes a smart meter installed onthe structure 338 that is interposed on the electric power line tomeasure or monitor and record electricity consumption. A conventionalpower meter measures electricity consumption, generally on a cumulativebasis, and the power utility periodically retrieves the cumulativeconsumption value from the power meter either through manual meterreading or, more routinely, through a wired or wireless interface thatis typically unidirectional and only allowing the retrieval of data.

A smart meter differs from a conventional power meter by allowing moregranular measurements of consumption, hourly or more frequently, andmore frequent reporting to the utility, whether on a daily basis or moreoften. Smart meters further differ from conventional power meters byoffering bidirectional communications capabilities with the powerutility, which is able to thereby offer the ability to reduce load,disconnect-reconnect service, and interface to other utility servicemeters, such as gas and water meters. Moreover, a power utility is ableto retrieve consumption data from a smart meter as needed for otherpurposes, including performing a Virtual Energy Audit analysis, asdescribed herein. Note that smart meters are also available formeasuring or monitoring other types of energy, fuel, and commodityconsumption or usage, including natural gas, liquid propane, and water.

The computer system 331 needs two or three sources of data, dependingupon whether on-site power generation 340 is installed. First, net load334 is required, which can be obtained from a power utility or fromother sources of data, such as a customer's utility bill records. Thenet load 334 could also be measured on-site, provided that the powermetering infrastructure 336 is available and accessible, such as througha smart meter. Where no on-site power generation is in use, the energyload 341 of the structure 338 is simply the net load 334, as verifiedand adjusted for stability, as described supra with reference to FIGS.17 and 24.

Second, outdoor temperature 335 for the location of the structure 338,as measured over the same period of time as the net load 334, isrequired. This data could be externally-supplied, such as through theSolarAnywhere® database service, cited supra, or other sources ofmeteorological data, or measured on-site, provided that outdoortemperature monitoring equipment 337 is available and accessible.

Third, where applicable, on-site power generation 340, as measured overthe same period of time, will be required in those situations whereon-site power generation 340 is either metered, typically by the samepower utility that supplies electricity, yet not combined with the netload 334, or there is no separate power meter installed for measuringon-site power generation 340. If a separate solar power meter isinstalled, the energy load 341 is simply the sum of the net load 334 andthe on-site power generation 340.

On-site power generation 340 will typically be supplied through solarpower generation 339, although other forms of on-site power generationmay apply, and must be added to the net load 334 to reconstruct theenergy load 341. However, if no solar power meter is installed, theon-site power generation 340 will need to be obtained from eitherhistorical records, provided that on-site power generation 340 wasmeasured and recorded (but not actually metered), or estimated byretrospectively “forecasting” power generation, for instance, by using aprobabilistic forecast of photovoltaic fleet power generation, such asdescribed supra beginning with reference to FIG. 1, by assuming a fleetsize that consists of only one system in Equation (12). The energy load341 is then calculated as the sum of the net load 334 and the historicalor estimated on-site power generation 340.

Based on these two or three sources of data, the computer system 331executes a software program 332 to analyze building performance usingthe Virtual Energy Audit methodology, described supra with reference toFIGS. 17 and 24. The resulting disaggregated component load results 333for the structure 338 can then be provided to the home or building owneror to the power utility for use in power generation operations andplanning and other purposes. For instance, a power utility could use theanalytical findings for assessing on-going and forecasted powerconsumption by the power utility's customer base, and in enabling thepower utility to adjust or modify the generation or procurement ofelectric power as a function of a power utility's remote consumer energyauditing analytical findings.

The computer system 331 includes hardware components found in a generalpurpose programmable computing device, such as a central processingunit, memory, input/output ports, network interface, and non-volatilestorage, and execute the software programs 332, as structured intoroutines, functions, and modules. In addition, other configurations ofcomputational resources, whether provided as a dedicated system orarranged in client-server or peer-to-peer topologies, and includingunitary or distributed processing, communications, storage, and userinterfacing, are possible.

Applications

Virtual Energy Audit Use Cases

The Virtual Energy Audit analysis methodology supports multiple usecases for the benefit of power utilities and their customers. Here aresome examples.

-   -   Provide building-specific, objective parameters that are useful        in modeling the building's energy consumption.    -   Produce the Effective R-Value, an intuitive comparative metric        of overall building thermal performance.    -   Disaggregate electrical loads into heating loads, cooling loads,        baseloads, and other loads.    -   Forecast heating and cooling energy consumption over a        multi-year horizon.    -   Infer space heating system technology type.    -   Predict natural gas heating consumption based only on electrical        usage data.    -   Calculate benefits of switching from natural gas to electric        heating.    -   Guide customers in building shell, HVAC, smart thermostat and        other energy investment decisions.    -   Detect when an AC system requires servicing.    -   Forecast short-term electrical building usage to support utility        load forecasting.    -   Quantify building performance according to known standards.    -   Support power utility programs that target customers with the        greatest potential savings opportunities.    -   Inform Distributed Energy Resource (DER) adoption models of        building thermal technologies.    -   Serve as a measurement and verification tool for building shell,        HVAC, and smart thermostat investments.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

The invention claimed is:
 1. A method for determining seasonal energyconsumption with the aid of a digital computer, comprising the steps of:assessing through a power meter energy loads for a building situated ina known location as measured over a seasonal time period; assessingoutdoor temperatures for the building as measured over the seasonal timeperiod through a temperature monitoring infrastructure; and operating adigital computer associated with a power utility and comprising aprocessor and a memory that is adapted to store program instructions forexecution by the processor, the program instructions capable of:expressing each energy load as a function of the outdoor temperaturemeasured at the same time of the seasonal time period in point-interceptform; and taking a slope of the point-intercept form as a fuel rate ofenergy consumption during the seasonal time period; and using the fuelrate of energy consumption for balancing power output of powergeneration equipment operated under control of the power utility.
 2. Amethod according to claim 1, further comprising: finding a balance pointtemperature for the seasonal time period as the outdoor temperatureabove which the fuel rate of energy consumption deviates from thepoint-intercept form.
 3. A method according to claim 1, furthercomprising: determining on-site power generation for the building asprovided at the same time of the seasonal time period; and adding theon-site power generation to each energy load prior to expressing eachenergy load as a function of the outdoor temperature.
 4. A methodaccording to claim 3, wherein the on-site power generation comprises aphotovoltaic system for the building, further comprising: generating aset of sky clearness indexes as a ratio of each irradiance observationin a set of irradiance observations that has been regularly measured forthe known location, and clear sky irradiance; forming a time series ofthe set of the sky clearness indexes; determining irradiance statisticsfor the photovoltaic system through statistical evaluation of the timeseries of the set of the sky clearness indexes; and building powerstatistics for the photovoltaic system as a function of the photovoltaicsystem irradiance statistics and an overall power rating of thephotovoltaic system.
 5. A method according to claim 1, furthercomprising at least one of: measuring on-site power generation atperiodic intervals; remotely measuring the energy load at periodicintervals through a power metering infrastructure; and measuring theenergy load on-site at periodic intervals through a further powermetering infrastructure.
 6. A method according to claim 1, furthercomprising at least one of: remotely measuring the outdoor temperatureat periodic intervals through a temperature monitoring infrastructure;and measuring the outdoor temperature on-site at periodic intervalsthrough a further temperature monitoring Infrastructure.
 7. A methodaccording to claim 1, further comprising at least one of: defining aheat transfer coefficient representing an overall resistance to heatflow through all surfaces comprising a surface area of the building as afunction of the surface area of the building, the fuel rate of energyconsumption for the seasonal time period requiring heating of thebuilding, and a heating efficiency of the heating system used in thebuilding; and defining a cool transfer coefficient representing anoverall resistance to cool flow through all surfaces comprising thesurface area of the building as a function of the surface area of thebuilding, the fuel rate of energy consumption for the seasonal timeperiod requiring cooling of the building, and a cooling efficiency ofthe cooling system used in the building.
 8. A method according to claim7, wherein the heat transfer coefficient R^(Effective) is determined inaccordance with:$R^{Effective} = \frac{A^{Total}}{{FuelRate}^{Heating}*\eta^{Heating}}$where A^(Total) represents the surface area of the building,FuelRate^(Heating) represents the fuel rate of energy consumption forthe seasonal time period requiring heating of the building, andn^(Heating) represents the heating efficiency of the heating system usedin the building.
 9. A method according to claim 7, wherein the cooltransfer coefficient R^(Effective) is determined in accordance with:$R^{Effective} = \frac{A^{Total}}{{FuelRate}^{Cooling}*\eta^{Cooling}}$where A^(Total) represents the surface area of the building,FuelRate^(Cooling) represents the fuel rate of energy consumption forthe seasonal time period requiring cooling of the building, andn^(Cooling) represents the cooling efficiency of the cooling system usedin the building.
 10. A method according to claim 7, further comprising:estimating the surface area of the building as a function of a floorarea and heights and number of floors of the building.
 11. A methodaccording to claim 10, wherein the surface area of the buildingA^(Total) is determined in accordance with:$A^{Total} = {{2\left( \frac{A}{F} \right)} + {4H\sqrt{AF}}}$ where Arepresents the floor area of a building, F represents the number offloors, and H represents the height per floor.
 12. A system fordetermining seasonal energy consumption with the aid of a digitalcomputer, comprising: a digital computer associated with a power utilityand comprising a processor and a memory that is adapted to store programinstructions for execution by the processor, the program instructionsconfigured to: assess through a power meter energy loads for a buildingsituated in a known location as measured over a seasonal time period;assess outdoor temperatures for the building as measured over theseasonal time period through a temperature monitoring infrastructure;express each energy load as a function of the outdoor temperaturemeasured at the same time of the seasonal time period in point-interceptform; take a slope of the point-intercept form as a fuel rate of energyconsumption during the seasonal time period; and use the fuel rate ofenergy consumption for balancing power output of power generationequipment operated under control of the power utility.
 13. A systemaccording to claim 12, the program instructions further configured to:find a balance point temperature for the seasonal time period as theoutdoor temperature above which the fuel rate of energy consumptiondeviates from the point-intercept form.
 14. A system according to claim12, the program instructions further configured to: determine on-sitepower generation for the building as provided at the same time of theseasonal time period; and add the on-site power generation to eachenergy load prior to expressing each energy load as a function of theoutdoor temperature.
 15. A system according to claim 14, wherein theon-site power generation comprises a photovoltaic system for thebuilding, the program instructions further configured to: generate a setof sky clearness indexes as a ratio of each irradiance observation in aset of irradiance observations that has been regularly measured for theknown location, and clear sky irradiance; form a time series of the setof the sky clearness indexes; determine irradiance statistics for thephotovoltaic system through statistical evaluation of the time series ofthe set of the sky clearness indexes; and build power statistics for thephotovoltaic system as a function of the photovoltaic system irradiancestatistics and an overall power rating of the photovoltaic system.
 16. Asystem according to claim 12, the program instructions furtherconfigured to: measure on-site power generation at periodic intervals;remotely measure the energy load at periodic intervals through a powermetering infrastructure; and measure the energy load on-site at periodicintervals through a further power metering infrastructure.
 17. A systemaccording to claim 12, the program instructions further configured to:remotely measure the outdoor temperature at periodic intervals through atemperature monitoring infrastructure, which is located off-site fromthe building; and measure the outdoor temperature on-site at periodicintervals through a further temperature monitoring infrastructure.
 18. Asystem according to claim 12, the program instructions furtherconfigured to at least one of: define a heat transfer coefficientrepresenting an overall resistance to heat flow through all surfacescomprising the surface area of the building as a function of the surfacearea of the building, the fuel rate of energy consumption for theseasonal time period requiring heating of the building, and a heatingefficiency of a heating system used in the building; and define a cooltransfer coefficient representing an overall resistance to cool flowthrough all surfaces comprising the surface area of the building as afunction of the surface area of the building, the fuel rate of energyconsumption for the seasonal time period requiring cooling of thebuilding, and a cooling efficiency of a cooling system used in thebuilding.
 19. A system according to claim 18, wherein the heat transfercoefficient R^(Effective) is determined in accordance with:$R^{Effective} = \frac{A^{Total}}{{FuelRate}^{Heating}*\eta^{Heating}}$where A^(Total) represents the surface area of the building,FuelRate^(Heating) represents the fuel rate of energy consumption forthe seasonal time period requiring heating of the building, andn^(Heating) represents the heating efficiency of the heating system usedin the building.
 20. A system according to claim 18, wherein the cooltransfer coefficient R^(Effective) is determined in accordance with:$R^{Effective} = \frac{A^{Total}}{{FuelRate}^{Cooling}*\eta^{Cooling}}$where A^(Total) represents the surface area of the building,FuelRate^(Cooling) represents the fuel rate of energy consumption forthe seasonal time period requiring cooling of the building, andn^(Cooling) represents the cooling efficiency of the cooling system usedin the building.